text: # -*- coding: utf-8 -*-
  r"""
  Finite word
  
  AUTHORS:
  
  - Arnaud Bergeron
  - Amy Glen
  - Sébastien Labbé
  - Franco Saliola
  - Julien Leroy (March 2010): reduced_rauzy_graph
  
  EXAMPLES:
  
  =========================
  Creation of a finite word
  =========================
  
  Finite words from Python strings, lists and tuples::
  
      sage: Word("abbabaab")
      word: abbabaab
      sage: Word([0, 1, 1, 0, 1, 0, 0, 1])
      word: 01101001
      sage: Word( ('a', 0, 5, 7, 'b', 9, 8) )
      word: a057b98
  
  Finite words from functions::
  
      sage: f = lambda n : n%3
      sage: Word(f, length=13)
      word: 0120120120120
  
  Finite words from iterators::
  
      sage: from itertools import count
      sage: Word(count(), length=10)
      word: 0123456789
  
  ::
  
      sage: Word( iter('abbccdef') )
      word: abbccdef
  
  Finite words from words via concatenation::
  
      sage: u = Word("abcccabba")
      sage: v = Word([0, 4, 8, 8, 3])
      sage: u * v
      word: abcccabba04883
      sage: v * u
      word: 04883abcccabba
      sage: u + v
      word: abcccabba04883
      sage: u^3 * v^(8/5)
      word: abcccabbaabcccabbaabcccabba04883048
  
  Finite words from infinite words::
  
      sage: vv = v^Infinity
      sage: vv[10000:10015]
      word: 048830488304883
  
  Finite words in a specific combinatorial class::
  
      sage: W = Words("ab")
      sage: W
      Finite and infinite words over {'a', 'b'}
      sage: W("abbabaab")
      word: abbabaab
      sage: W(["a","b","b","a","b","a","a","b"])
      word: abbabaab
      sage: W( iter('ababab') )
      word: ababab
  
  Finite word as the image under a morphism::
  
      sage: m = WordMorphism({0:[4,4,5,0],5:[0,5,5],4:[4,0,0,0]})
      sage: m(0)
      word: 4450
      sage: m(0, order=2)
      word: 400040000554450
      sage: m(0, order=3)
      word: 4000445044504450400044504450445044500550...
  
  .. NOTE::
  
      The following two finite words have the same string representation::
  
          sage: w = Word('010120')
          sage: z = Word([0, 1, 0, 1, 2, 0])
          sage: w
          word: 010120
          sage: z
          word: 010120
  
      but are not equal::
  
          sage: w == z
          False
  
      Indeed, w and z are defined on different alphabets::
  
          sage: w[2]
          '0'
          sage: z[2]
          0
  
  ========================
  Functions and algorithms
  ========================
  
  There are more than 100 functions defined on a finite word. Here are some
  of them::
  
      sage: w = Word('abaabbba'); w
      word: abaabbba
      sage: w.is_palindrome()
      False
      sage: w.is_lyndon()
      False
      sage: w.number_of_factors()
      28
      sage: w.critical_exponent()
      3
  
  ::
  
      sage: print(w.lyndon_factorization())
      (ab, aabbb, a)
      sage: print(w.crochemore_factorization())
      (a, b, a, ab, bb, a)
  
  ::
  
      sage: st = w.suffix_tree()
      sage: st
      Implicit Suffix Tree of the word: abaabbba
      sage: st.show(word_labels=True)
  
  ::
  
      sage: T = words.FibonacciWord('ab')
      sage: T.longest_common_prefix(Word('abaabababbbbbb'))
      word: abaababa
  
  As matrix and many other sage objects, words have a parent::
  
      sage: u = Word('xyxxyxyyy')
      sage: u.parent()
      Finite words over Set of Python objects of class 'object'
  
  ::
  
      sage: v = Word('xyxxyxyyy', alphabet='xy')
      sage: v.parent()
      Finite words over {'x', 'y'}
  
  ========================
  Factors and Rauzy Graphs
  ========================
  
  Enumeration of factors, the successive values returned by ``next(it)``
  can appear in a different order depending on hardware. Therefore we
  mark the three first results of the test ``random``. The important test
  is that the iteration stops properly on the fourth call::
  
      sage: w = Word([4,5,6])^7
      sage: it = w.factor_iterator(4)
      sage: next(it) # random
      word: 6456
      sage: next(it) # random
      word: 5645
      sage: next(it) # random
      word: 4564
      sage: next(it)
      Traceback (most recent call last):
      ...
      StopIteration
  
  The set of factors::
  
      sage: sorted(w.factor_set(3))
      [word: 456, word: 564, word: 645]
      sage: sorted(w.factor_set(4))
      [word: 4564, word: 5645, word: 6456]
      sage: w.factor_set().cardinality()
      61
  
  Rauzy graphs::
  
      sage: f = words.FibonacciWord()[:30]
      sage: f.rauzy_graph(4)
      Looped digraph on 5 vertices
      sage: f.reduced_rauzy_graph(4)
      Looped multi-digraph on 2 vertices
  
  Left-special and bispecial factors::
  
      sage: f.number_of_left_special_factors(7)
      1
      sage: f.bispecial_factors()
      [word: , word: 0, word: 010, word: 010010, word: 01001010010]
  """
  #*****************************************************************************
  #       Copyright (C) 2008 Arnaud Bergeron <abergeron@gmail.com>,
  #                     2008 Amy Glen <amy.glen@gmail.com>,
  #                     2008-2012 Sébastien Labbé <slabqc@gmail.com>,
  #                     2008-2010 Franco Saliola <saliola@gmail.com>
  #
  # This program is free software: you can redistribute it and/or modify
  # it under the terms of the GNU General Public License as published by
  # the Free Software Foundation, either version 2 of the License, or
  # (at your option) any later version.
  #                  http://www.gnu.org/licenses/
  #*****************************************************************************
  from __future__ import print_function, absolute_import
  
  from builtins import zip
  
  from six.moves import range
  from collections import defaultdict
  from itertools import islice, cycle
  from sage.combinat.words.abstract_word import Word_class
  from sage.combinat.words.words import Words
  from sage.misc.cachefunc import cached_method
  from sage.combinat.words.word_options import word_options
  from sage.rings.all import Integer, Infinity, ZZ
  from sage.sets.set import Set
  from sage.misc.superseded import deprecated_function_alias
  
  class FiniteWord_class(Word_class):
      def __str__(self):
          r"""
          Return the full (not truncated) string representation of the word
          without identifier.
  
          TESTS::
  
              sage: Word('abc').__str__()
              'abc'
              sage: Word([0, 1, 0, 0, 1] * 10).__str__()
              '01001010010100101001010010100101001010010100101001'
              sage: Word([0,1,10,101]).__str__()
              '0,1,10,101'
  
          Insertion into a ``str``::
  
              sage: w = Word(range(5))
              sage: "Let's insert the word w = %s in this string." % w
              "Let's insert the word w = 01234 in this string."
  
          Using ``LatexExpr``::
  
              sage: from sage.misc.latex import LatexExpr
              sage: LatexExpr(w)
              01234
  
          With the ``print`` statement::
  
              sage: print(w)
              01234
  
          No truncation is done for finite words::
  
              sage: w = Word([i % 5 for i in range(60)])
              sage: print(w)
              012340123401234012340123401234012340123401234012340123401234
          """
          global word_options
          if word_options['display'] == 'string':
              ls = word_options['letter_separator']
              letters = [str(_) for _ in self]
              if all(len(a)==1 for a in letters):
                  return ''.join(letters)
              else:
                  return ls.join(letters)
          elif word_options['display'] == 'list':
              return str(list(self))
  
      def _repr_(self):
          r"""
          Return a string representation of ``self``.
  
          TESTS::
  
              sage: Word(range(10))._repr_()
              'word: 0123456789'
              sage: Word(range(100))._repr_()
              'word: 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,...'
          """
          global word_options
          if word_options['old_repr']:
              if word_options['truncate'] and \
                      self.length() > word_options['truncate_length']:
                  return "Finite word of length %s over %s" % (self.length(), str(self.parent().alphabet())[17:])
          return word_options['identifier'] + self.string_rep()
  
      def coerce(self, other):
          r"""
          Try to return a pair of words with a common parent; raise an
          exception if this is not possible.
  
          This function begins by checking if both words have the same
          parent. If this is the case, then no work is done and both words
          are returned as-is.
  
          Otherwise it will attempt to convert ``other`` to the domain of ``self``.
          If that fails, it will attempt to convert ``self`` to the domain of
          ``other``. If both attempts fail, it raises a ``TypeError`` to signal
          failure.
  
          EXAMPLES::
  
              sage: W1 = Words('abc'); W2 = Words('ab')
              sage: w1 = W1('abc'); w2 = W2('abba'); w3 = W1('baab')
              sage: w1.parent() is w2.parent()
              False
              sage: a, b = w1.coerce(w2)
              sage: a.parent() is b.parent()
              True
              sage: w1.parent() is w2.parent()
              False
          """
          if self.parent() != other.parent():
              try:
                  other = self.parent()(other)
                  other.parent()._check(other, length=None)
              except Exception:
                  try:
                      self = other.parent()(self)
                      self.parent()._check(self, length=None)
                  except Exception:
                      raise TypeError("no coercion rule between %r and %r" % (self.parent(), other.parent()))
          return self, other
  
      def __hash__(self):
          r"""
          Return the hash for this word.
  
          TESTS::
  
               sage: h = hash(Word('abc'))    # indirect test
               sage: Word('abc').__hash__() == Word('abc').__hash__()
               True
          """
          if self._hash is None:
              res = 5381
              for s in self._to_integer_iterator():
                  res = ((res << 5) + res) + s
              self._hash = res
          return self._hash
  
      def concatenate(self, other):
          r"""
          Return the concatenation of ``self`` and ``other``.
  
          INPUT:
  
          - ``other`` -- a word over the same alphabet as ``self``
  
          EXAMPLES:
  
          Concatenation may be made using ``+`` or ``*`` operations::
  
              sage: w = Word('abadafd')
              sage: y = Word([5,3,5,8,7])
              sage: w * y
              word: abadafd53587
              sage: w + y
              word: abadafd53587
              sage: w.concatenate(y)
              word: abadafd53587
  
          Both words must be defined over the same alphabet::
  
              sage: z = Word('12223', alphabet = '123')
              sage: z + y
              Traceback (most recent call last):
              ...
              ValueError: 5 not in alphabet!
  
          Eventually, it should work::
  
              sage: z = Word('12223', alphabet = '123')
              sage: z + y                   #todo: not implemented
              word: 1222353587
  
          TESTS:
  
          The empty word is not considered by concatenation::
  
              sage: type(Word([]) * Word('abcd'))
              <class 'sage.combinat.words.word.FiniteWord_str'>
              sage: type(Word('abcd') * Word())
              <class 'sage.combinat.words.word.FiniteWord_str'>
              sage: type(Word('abcd') * Word([]))
              <class 'sage.combinat.words.word.FiniteWord_str'>
              sage: type(Word('abcd') * Word(()))
              <class 'sage.combinat.words.word.FiniteWord_str'>
              sage: type(Word([1,2,3]) * Word(''))
              <class 'sage.combinat.words.word.FiniteWord_list'>
  
          Concatenation of finite words with infinite words works as expected::
  
              sage: from itertools import repeat
              sage: W = Words('ab')
              sage: w1 = W('aba')
              sage: w2 = W(repeat('b'), length='infinite')
              sage: w1*w2
              word: ababbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb...
              sage: _.parent()
              Infinite words over {'a', 'b'}
          """
          if self.is_empty():
              return other
          if isinstance(other, Word_class) and other.is_empty():
              return self
          f = CallableFromListOfWords([self,other])
          length = self.length() + other.length()
          parent = self._parent
          if length == Infinity:
              parent = parent.shift()
              return parent(f, datatype='callable', caching=True)
          else:
              return parent(f, length=length, datatype='callable', caching=True)
  
      __mul__ = concatenate
  
      __add__ = concatenate
  
      # TODO: This function is using domain=range(n) for Word but
      # should be a domain=slice(n) # Seb : Feb 23th : I think this is fine now!!
      def __pow__(self, exp):
          r"""
          Return the ``exp``-th power of ``self``.
  
          If ``exp`` is `\infty`, returns the infinite periodic word of base ``self``.
          Otherwise, `|w|\cdot exp` must be a non-negative integer.
  
          INPUT:
  
          - ``exp`` -- an integer, a rational, a float number or plus infinity
  
          OUTPUT:
  
          word -- the ``exp``-th power of ``self``
  
          EXAMPLES:
  
          You can take non-negative integer powers::
  
              sage: w = Word(range(6)); w
              word: 012345
              sage: w^2
              word: 012345012345
              sage: w^1
              word: 012345
              sage: w^0
              word:
              sage: w^(-1)
              Traceback (most recent call last):
              ...
              ValueError: Power of the word is not defined on the exponent -1: the length of the word (6) times the exponent (-1) must be a positive integer
  
  
          You can take non-negative rational powers::
  
              sage: w = Word(range(6)); w
              word: 012345
              sage: w^(.5)
              word: 012
              sage: w^(1/3)
              word: 01
              sage: (w*w)^(1/2) == w
              True
              sage: w^(5/2)
              word: 012345012345012
  
          ...but the length of the word times the exponent must be an integer::
  
              sage: w = Word(range(6))
              sage: w^(1/4)
              Traceback (most recent call last):
              ...
              ValueError: Power of the word is not defined on the exponent 1/4: the length of the word (6) times the exponent (1/4) must be a positive integer
  
          You can take infinite power::
  
              sage: w = Word(range(6)); w
              word: 012345
              sage: u = w^oo; u
              word: 0123450123450123450123450123450123450123...
              sage: u[10000000:20000000]
              word: 4501234501234501234501234501234501234501...
              sage: u[10000000:10000020]
              word: 45012345012345012345
              sage: Word()^oo
              word:
          """
          # powers of the empty word
          if self.is_empty():
              return self
  
          # infinite power of a non-empty word
          fcn = lambda n: self[n % self.length()]
          if exp is Infinity:
              return self._parent.shift()(fcn)
  
          #If exp*|self| is not an integer
          length = exp* self.length()
          if length in ZZ and length >= 0:
              return self._parent(fcn, length=length)
          else:
              raise ValueError("Power of the word is not defined on the exponent {}:"
                      " the length of the word ({}) times the exponent ({}) must"
                      " be a positive integer".format(exp,self.length(),exp))
  
      def length(self):
          r"""
          Return the length of ``self``.
  
          TESTS::
  
              sage: from sage.combinat.words.word import Word_class
              sage: w = Word(iter('abba'*40), length="finite")
              sage: w._len is None
              True
              sage: w.length()
              160
              sage: w = Word(iter('abba'), length=4)
              sage: w._len
              4
              sage: w.length()
              4
              sage: def f(n):
              ....:   return list(range(2,12,2))[n]
              sage: w = Word(f, length=5)
              sage: w.length()
              5
          """
          if self._len is None:
              self._len = Integer(sum(1 for _ in self))
          return self._len
  
      def content(self, n=None):
          r"""
          Return content of ``self``.
  
          INPUT:
  
          - ``n`` -- (optional) an integer specifying the maximal
            letter in the alphabet
  
          OUTPUT:
  
          - a list where the `i`-th entry indiciates the multiplicity
            of the `i`-th letter in the alphabet in ``self``
  
          EXAMPLES::
  
              sage: w = Word([1,2,4,3,2,2,2])
              sage: w.content()
              [1, 4, 1, 1]
              sage: w = Word([3,1])
              sage: w.content()
              [1, 1]
              sage: w.content(n=3)
              [1, 0, 1]
              sage: w = Word([2,4],alphabet=[1,2,3,4])
              sage: w.content(n=3)
              [0, 1, 0]
              sage: w.content()
              [0, 1, 0, 1]
          """
          if n is not None:
              alphabet = range(1,n+1)
          elif not self.parent().alphabet().cardinality() == +Infinity:
              alphabet = self.parent().alphabet()
          else:
              alphabet = sorted(self.letters())
          return [self.count(i) for i in alphabet]
  
      def is_yamanouchi(self, n=None):
          r"""
          Return whether ``self`` is Yamanouchi.
  
          A word `w` is Yamanouchi if, when read from right to left, it
          always has weakly more `i`'s than `i+1`'s for all `i` that
          appear in `w`.
  
          INPUT:
  
          - ``n`` -- (optional) an integer specifying the maximal
            letter in the alphabet
  
          EXAMPLES::
  
              sage: w = Word([1,2,4,3,2,2,2])
              sage: w.is_yamanouchi()
              False
              sage: w = Word([2,3,4,3,1,2,1,1,2,1])
              sage: w.is_yamanouchi()
              True
              sage: w = Word([3,1])
              sage: w.is_yamanouchi(n=3)
              False
              sage: w.is_yamanouchi()
              True
              sage: w = Word([3,1],alphabet=[1,2,3])
              sage: w.is_yamanouchi()
              False
              sage: w = Word([2,1,1,2])
              sage: w.is_yamanouchi()
              False
          """
          from sage.combinat.words.word import Word
          if n is not None:
              w = Word(self, alphabet=list(range(1,n+1)))
          elif not self.parent().alphabet().cardinality() == +Infinity:
              w = self
          else:
              w = Word(self, alphabet=sorted(self.letters()))
          l = w.length()
          for a in range(l-1,-1,-1):
              mu = w.parent()(self[a:]).content()
              if not all(mu[i] >= mu[i+1] for i in range(len(mu)-1)):
                  return False
          return True
  
      def schuetzenberger_involution(self, n = None):
          """
          Return the Schützenberger involution of the word ``self``, which is obtained
          by reverting the word and then complementing all letters within the
          underlying ordered alphabet. If ``n`` is specified, the underlying
          alphabet is assumed to be `[1,2,\ldots,n]`. If no alphabet is specified,
          `n` is the maximal letter appearing in ``self``.
  
          INPUT:
  
          - ``self`` -- a word
          - ``n`` -- an integer specifying the maximal letter in the alphabet (optional)
  
          OUTPUT:
  
          a word, the Schützenberger involution of ``self``
  
          EXAMPLES::
  
              sage: w = Word([9,7,4,1,6,2,3])
              sage: v = w.schuetzenberger_involution(); v
              word: 7849631
              sage: v.parent()
              Finite words over Set of Python objects of class 'object'
  
              sage: w = Word([1,2,3],alphabet=[1,2,3,4,5])
              sage: v = w.schuetzenberger_involution();v
              word: 345
              sage: v.parent()
              Finite words over {1, 2, 3, 4, 5}
  
              sage: w = Word([1,2,3])
              sage: v = w.schuetzenberger_involution(n=5);v
              word: 345
              sage: v.parent()
              Finite words over Set of Python objects of class 'object'
  
              sage: w = Word([11,32,69,2,53,1,2,3,18,41])
              sage: w.schuetzenberger_involution()
              word: 29,52,67,68,69,17,68,1,38,59
  
              sage: w = Word([],alphabet=[1,2,3,4,5])
              sage: w.schuetzenberger_involution()
              word:
  
              sage: w = Word([])
              sage: w.schuetzenberger_involution()
              word:
          """
          if self.length() == 0:
              return self
          r = self.reversal()
          w = list(r)
          parent = self.parent()
          if n is None:
              alphsize = parent.alphabet().cardinality()
              if not alphsize == +Infinity:
                  n = max(parent.alphabet())
              elif r.length()>0:
                  n = max(w)
          for k in range(r.length()):
              w[k] = n+1 - w[k]
          return parent(w, check=False)
  
      def foata_bijection(self):
          r"""
          Return word ``self`` under the Foata bijection.
  
          The Foata bijection `\phi` is a bijection on the set of words
          of given content (by a slight generalization of Section 2 in [FoSc78]_).
          It can be defined by induction on the size of the word: Given a word
          `w_1 w_2 \cdots w_n`, start with `\phi(w_1) = w_1`. At the `i`-th step, if
          `\phi(w_1 w_2 \cdots w_i) = v_1 v_2 \cdots v_i`, we define
          `\phi(w_1 w_2 \cdots w_i w_{i+1})` by placing `w_{i+1}` on the end of
          the word `v_1 v_2 \cdots v_i` and breaking the word up into blocks
          as follows. If `w_{i+1} \ge v_i`, place a vertical line to the right
          of each `v_k` for which `w_{i+1} \ge v_k`. Otherwise, if
          `w_{i+1} < v_i`, place a vertical line to the right of each `v_k`
          for which `w_{i+1} < v_k`. In either case, place a vertical line at
          the start of the word as well. Now, within each block between
          vertical lines, cyclically shift the entries one place to the
          right.
  
          For instance, to compute `\phi([4,1,5,4,2,2,3])`, the sequence of
          words is
  
          * `4`,
          * `|4|1 \to 41`,
          * `|4|1|5 \to 415`,
          * `|415|4 \to 5414`,
          * `|5|4|14|2 \to 54412`,
          * `|5441|2|2 \to 154422`,
          * `|1|5442|2|3 \to 1254423`.
  
          So `\phi([4,1,5,4,2,2,3]) = [1,2,5,4,4,2,3]`.
  
          .. SEEALSO::
  
              :meth:`Foata bijection on Permutations <sage.combinat.permutation.Permutation.foata_bijection()>`.
  
          EXAMPLES::
  
              sage: w = Word([2,2,2,1,1,1])
              sage: w.foata_bijection()
              word: 112221
              sage: w = Word([2,2,1,2,2,2,1,1,2,1])
              sage: w.foata_bijection()
              word: 2122212211
              sage: w = Word([4,1,5,4,2,2,3])
              sage: w.foata_bijection()
              word: 1254423
  
          TESTS::
  
              sage: w = Word('121314')
              sage: w.foata_bijection()
              word: 231114
              sage: w = Word('1133a1')
              sage: w.foata_bijection()
              word: 3113a1
          """
          s = self.standard_permutation()
          ordered_alphabet = sorted(self.letters(),
                                    key=self.parent().sortkey_letters)
          eval_dict = self.evaluation_dict()
          weight = [eval_dict[a] for a in ordered_alphabet]
          return (s.foata_bijection()).destandardize(weight, ordered_alphabet=ordered_alphabet)
  
      def major_index(self, final_descent=False):
          r"""
          Return the major index of ``self``.
  
          The major index of a word `w` is the sum of the descents of `w`.
  
          With the ``final_descent`` option, the last position of a
          non-empty word is also considered as a descent.
  
          .. SEEALSO::
  
              :meth:`major index on Permutations <sage.combinat.permutation.Permutation.major_index()>`.
  
          EXAMPLES::
  
              sage: w = Word([2,1,3,3,2])
              sage: w.major_index()
              5
              sage: w = Word([2,1,3,3,2])
              sage: w.major_index(final_descent=True)
              10
          """
          return (self.standard_permutation()).major_index(final_descent=final_descent)
  
      def number_of_inversions(self):
          r"""
          Return the number of inversions in ``self``.
  
          An inversion of a word `w = w_1 \ldots w_n` is a pair of indices `(i, j)`
          with `i < j` and `w_i > w_j`.
  
          .. SEEALSO::
  
              :meth:`number of inversions on Permutations <sage.combinat.permutation.Permutation.number_of_inversions()>`.
  
          EXAMPLES::
  
              sage: w = Word([2,1,3,3,2])
              sage: w.number_of_inversions()
              3
          """
          return (self.standard_permutation()).number_of_inversions()
  
      def is_empty(self):
          r"""
          Return ``True`` if the length of ``self`` is zero, and ``False`` otherwise.
  
          EXAMPLES::
  
              sage: Word([]).is_empty()
              True
              sage: Word('a').is_empty()
              False
          """
          return self.length()==0
  
      def is_finite(self):
          r"""
          Return ``True``.
  
          EXAMPLES::
  
              sage: Word([]).is_finite()
              True
              sage: Word('a').is_finite()
              True
          """
          return True
  
      def to_integer_word(self):
          r"""
          Return a word defined over the integers ``[0,1,...,self.length()-1]``
          whose letters are in the same relative order in the parent.
  
          EXAMPLES::
  
              sage: from itertools import count
              sage: w = Word('abbabaab')
              sage: w.to_integer_word()
              word: 01101001
              sage: w = Word(iter("cacao"), length="finite")
              sage: w.to_integer_word()
              word: 10102
  
              sage: w = Words([3,2,1])([2,3,3,1])
              sage: w.to_integer_word()
              word: 1002
          """
          from sage.combinat.words.word import Word
          return Word(self.to_integer_list())
  
      def to_integer_list(self):
          r"""
          Return a list of integers from ``[0,1,...,self.length()-1]`` in the
          same relative order as the letters in ``self`` in the parent.
  
          EXAMPLES::
  
              sage: from itertools import count
              sage: w = Word('abbabaab')
              sage: w.to_integer_list()
              [0, 1, 1, 0, 1, 0, 0, 1]
              sage: w = Word(iter("cacao"), length="finite")
              sage: w.to_integer_list()
              [1, 0, 1, 0, 2]
              sage: w = Words([3,2,1])([2,3,3,1])
              sage: w.to_integer_list()
              [1, 0, 0, 2]
          """
          cmp_key = self._parent.sortkey_letters
          ordered_alphabet = sorted(self.letters(), key=cmp_key)
          index = dict((b,a) for (a,b) in enumerate(ordered_alphabet))
          return [index[a] for a in self]
  
      # To fix : do not slice here ! (quite expensive in copy)
      def is_suffix(self, other):
          r"""
          Return ``True`` if ``self`` is a suffix of ``other``, and ``False`` otherwise.
  
          EXAMPLES::
  
              sage: w = Word('0123456789')
              sage: y = Word('56789')
              sage: y.is_suffix(w)
              True
              sage: w.is_suffix(y)
              False
              sage: Word('579').is_suffix(w)
              False
              sage: Word().is_suffix(y)
              True
              sage: w.is_suffix(Word())
              False
              sage: Word().is_suffix(Word())
              True
          """
          return self.is_empty() or self == other[-self.length():]
  
      def is_proper_suffix(self, other):
          r"""
          Return ``True`` if ``self`` is a proper suffix of ``other``, and ``False`` otherwise.
  
          EXAMPLES::
  
              sage: Word('23').is_proper_suffix(Word('123'))
              True
              sage: Word('12').is_proper_suffix(Word('12'))
              False
              sage: Word().is_proper_suffix(Word('123'))
              True
              sage: Word('123').is_proper_suffix(Word('12'))
              False
          """
          return self.is_suffix(other) and self.length() < other.length()
  
      def has_suffix(self, other):
          """
          Test whether ``self`` has ``other`` as a suffix.
  
          .. note::
  
             Some word datatype classes, like :class:`WordDatatype_str`,
             override this method.
  
          INPUT:
  
          - ``other`` -- a word, or data describing a word
  
          OUTPUT:
  
          boolean
  
          EXAMPLES::
  
              sage: w = Word("abbabaabababa")
              sage: u = Word("ababa")
              sage: w.has_suffix(u)
              True
              sage: u.has_suffix(w)
              False
              sage: u.has_suffix("ababa")
              True
  
          ::
  
              sage: w = Word([0,1,1,0,1,0,0,1,0,1,0,1,0])
              sage: u = Word([0,1,0,1,0])
              sage: w.has_suffix(u)
              True
              sage: u.has_suffix(w)
              False
              sage: u.has_suffix([0,1,0,1,0])
              True
  
          """
          from sage.combinat.words.word import Word
          w = Word(other)
          return w.is_suffix(self)
  
      def is_prefix(self, other):
          r"""
          Return ``True`` if ``self`` is a prefix of ``other``, and ``False`` otherwise.
  
          EXAMPLES::
  
              sage: w = Word('0123456789')
              sage: y = Word('012345')
              sage: y.is_prefix(w)
              True
              sage: w.is_prefix(y)
              False
              sage: w.is_prefix(Word())
              False
              sage: Word().is_prefix(w)
              True
              sage: Word().is_prefix(Word())
              True
          """
          return self == other[:self.length()]
  
      def is_proper_prefix(self, other):
          r"""
          Return ``True`` if ``self`` is a proper prefix of ``other``, and ``False`` otherwise.
  
          EXAMPLES::
  
              sage: Word('12').is_proper_prefix(Word('123'))
              True
              sage: Word('12').is_proper_prefix(Word('12'))
              False
              sage: Word().is_proper_prefix(Word('123'))
              True
              sage: Word('123').is_proper_prefix(Word('12'))
              False
              sage: Word().is_proper_prefix(Word())
              False
          """
          return self.is_prefix(other) and self.length() < other.length()
  
      def has_prefix(self, other):
          r"""
          Test whether ``self`` has ``other`` as a prefix.
  
          INPUT:
  
          - ``other`` -- a word, or data describing a word
  
          OUTPUT:
  
          boolean
  
          EXAMPLES::
  
              sage: w = Word("abbabaabababa")
              sage: u = Word("abbab")
              sage: w.has_prefix(u)
              True
              sage: u.has_prefix(w)
              False
              sage: u.has_prefix("abbab")
              True
  
          ::
  
              sage: w = Word([0,1,1,0,1,0,0,1,0,1,0,1,0])
              sage: u = Word([0,1,1,0,1])
              sage: w.has_prefix(u)
              True
              sage: u.has_prefix(w)
              False
              sage: u.has_prefix([0,1,1,0,1])
              True
  
  
          """
          from sage.combinat.words.word import Word
          w = Word(other)
          return w.is_prefix(self)
  
      def reversal(self):
          r"""
          Return the reversal of ``self``.
  
          EXAMPLES::
  
              sage: Word('124563').reversal()
              word: 365421
          """
          return self[::-1]
  
      @cached_method
      def prefix_function_table(self):
          r"""
          Return a vector containing the length of the proper prefix-suffixes
          for all the non-empty prefixes of ``self``.
  
          EXAMPLES::
  
              sage: Word('121321').prefix_function_table()
              [0, 0, 1, 0, 0, 1]
              sage: Word('1241245').prefix_function_table()
              [0, 0, 0, 1, 2, 3, 0]
              sage: Word().prefix_function_table()
              []
          """
          k = 0
          res = [0]*self.length()
          for q in range(1, self.length()):
              while k > 0 and self[k] != self[q]:
                  k = res[k-1]
              if self[k] == self[q]:
                  k += 1
              res[q] = k
          return res
  
      @cached_method
      def good_suffix_table(self):
          r"""
          Return a table of the maximum skip you can do in order not to miss
          a possible occurrence of ``self`` in a word.
  
          This is a part of the Boyer-Moore algorithm to find factors. See [1].
  
          EXAMPLES::
  
              sage: Word('121321').good_suffix_table()
              [5, 5, 5, 5, 3, 3, 1]
              sage: Word('12412').good_suffix_table()
              [3, 3, 3, 3, 3, 1]
  
          REFERENCES:
  
          -   [1] R.S. Boyer, J.S. Moore, A fast string searching algorithm,
              Communications of the ACM 20 (1977) 762--772.
          """
          l = self.length()
          p = self.reversal().prefix_function_table()
          res = [l - p[-1]]*(l+1)
          for i in range(1, l+1):
              j = l - p[i - 1]
              if res[j] > (i - p[i-1]):
                  res[j] = i - p[i-1]
          return res
  
      @cached_method
      def suffix_trie(self):
          r"""
          Return the suffix trie of ``self``.
  
          The *suffix trie* of a finite word `w` is a data structure
          representing the factors of `w`. It is a tree whose edges are
          labelled with letters of `w`, and whose leafs correspond to
          suffixes of `w`.
  
          Type ``sage.combinat.words.suffix_trees.SuffixTrie?`` for more information.
  
          EXAMPLES::
  
              sage: w = Word("cacao")
              sage: w.suffix_trie()
              Suffix Trie of the word: cacao
  
          ::
  
              sage: w = Word([0,1,0,1,1])
              sage: w.suffix_trie()
              Suffix Trie of the word: 01011
          """
          from sage.combinat.words.suffix_trees import SuffixTrie
          return SuffixTrie(self)
  
      def implicit_suffix_tree(self):
          r"""
          Return the implicit suffix tree of ``self``.
  
          The *suffix tree* of a word `w` is a compactification of the
          suffix trie for `w`. The compactification removes all nodes that have
          exactly one incoming edge and exactly one outgoing edge. It consists of
          two components: a tree and a word. Thus, instead of labelling the edges
          by factors of `w`, we can label them by indices of the occurrence of
          the factors in `w`.
  
          Type ``sage.combinat.words.suffix_trees.ImplicitSuffixTree?`` for more information.
  
          EXAMPLES::
  
              sage: w = Word("cacao")
              sage: w.implicit_suffix_tree()
              Implicit Suffix Tree of the word: cacao
  
          ::
  
              sage: w = Word([0,1,0,1,1])
              sage: w.implicit_suffix_tree()
              Implicit Suffix Tree of the word: 01011
          """
          from sage.combinat.words.suffix_trees import ImplicitSuffixTree
          return ImplicitSuffixTree(self)
  
      @cached_method
      def suffix_tree(self):
          r"""
          Alias for ``implicit_suffix_tree()``.
  
          EXAMPLES::
  
              sage: Word('abbabaab').suffix_tree()
              Implicit Suffix Tree of the word: abbabaab
          """
          return self.implicit_suffix_tree()
  
      def number_of_factors(self, n=None, algorithm='suffix tree'):
          r"""
          Count the number of distinct factors of ``self``.
  
          INPUT:
  
          -  ``n`` -- an integer, or ``None``.
          - ``algorithm`` -- string (default: ``'suffix tree'``), takes the
            following values:
  
            - ``'suffix tree'`` -- construct and use the suffix tree of the word
            - ``'naive'`` -- algorithm uses a sliding window
  
          OUTPUT:
  
          If ``n`` is an integer, returns the number of distinct factors
          of length ``n``. If ``n`` is ``None``, returns the total number of
          distinct factors.
  
          EXAMPLES::
  
              sage: w = Word([1,2,1,2,3])
              sage: w.number_of_factors()
              13
              sage: [w.number_of_factors(i) for i in range(6)]
              [1, 3, 3, 3, 2, 1]
  
          ::
  
              sage: w = words.ThueMorseWord()[:100]
              sage: [w.number_of_factors(i) for i in range(10)]
              [1, 2, 4, 6, 10, 12, 16, 20, 22, 24]
  
          ::
  
              sage: Word('1213121').number_of_factors()
              22
              sage: Word('1213121').number_of_factors(1)
              3
  
          ::
  
              sage: Word('a'*100).number_of_factors()
              101
              sage: Word('a'*100).number_of_factors(77)
              1
  
          ::
  
              sage: Word().number_of_factors()
              1
              sage: Word().number_of_factors(17)
              0
  
          ::
  
              sage: blueberry = Word("blueberry")
              sage: blueberry.number_of_factors()
              43
              sage: [blueberry.number_of_factors(i) for i in range(10)]
              [1, 6, 8, 7, 6, 5, 4, 3, 2, 1]
          """
          if algorithm == 'suffix tree':
              return self.suffix_tree().number_of_factors(n)
          elif algorithm == 'naive':
              return len(self.factor_set(n, algorithm='naive'))
          else:
              raise ValueError('Unknown algorithm (={})'.format(algorithm))
  
      def factor_iterator(self, n=None):
          r"""
          Generate distinct factors of ``self``.
  
          INPUT:
  
          - ``n`` -- an integer, or ``None``.
  
          OUTPUT:
  
          If ``n`` is an integer, returns an iterator over all distinct
          factors of length ``n``. If ``n`` is ``None``, returns an iterator
          generating all distinct factors.
  
          EXAMPLES::
  
              sage: w = Word('1213121')
              sage: sorted( w.factor_iterator(0) )
              [word: ]
              sage: sorted( w.factor_iterator(10) )
              []
              sage: sorted( w.factor_iterator(1) )
              [word: 1, word: 2, word: 3]
              sage: sorted( w.factor_iterator(4) )
              [word: 1213, word: 1312, word: 2131, word: 3121]
              sage: sorted( w.factor_iterator() )
              [word: , word: 1, word: 12, word: 121, word: 1213, word: 12131, word: 121312, word: 1213121, word: 13, word: 131, word: 1312, word: 13121, word: 2, word: 21, word: 213, word: 2131, word: 21312, word: 213121, word: 3, word: 31, word: 312, word: 3121]
  
          ::
  
              sage: u = Word([1,2,1,2,3])
              sage: sorted( u.factor_iterator(0) )
              [word: ]
              sage: sorted( u.factor_iterator(10) )
              []
              sage: sorted( u.factor_iterator(1) )
              [word: 1, word: 2, word: 3]
              sage: sorted( u.factor_iterator(5) )
              [word: 12123]
              sage: sorted( u.factor_iterator() )
              [word: , word: 1, word: 12, word: 121, word: 1212, word: 12123, word: 123, word: 2, word: 21, word: 212, word: 2123, word: 23, word: 3]
  
          ::
  
              sage: xxx = Word("xxx")
              sage: sorted( xxx.factor_iterator(0) )
              [word: ]
              sage: sorted( xxx.factor_iterator(4) )
              []
              sage: sorted( xxx.factor_iterator(2) )
              [word: xx]
              sage: sorted( xxx.factor_iterator() )
              [word: , word: x, word: xx, word: xxx]
  
          ::
  
              sage: e = Word()
              sage: sorted( e.factor_iterator(0) )
              [word: ]
              sage: sorted( e.factor_iterator(17) )
              []
              sage: sorted( e.factor_iterator() )
              [word: ]
  
          TESTS::
  
              sage: type( Word('cacao').factor_iterator() )
              <... 'generator'>
          """
          return self.suffix_tree().factor_iterator(n)
  
      def factor_set(self, n=None, algorithm='suffix tree'):
          r"""
          Return the set of factors (of length ``n``) of ``self``.
  
          INPUT:
  
          - ``n`` -- an integer or ``None`` (default: ``None``).
          - ``algorithm`` -- string (default: ``'suffix tree'``), takes the
            following values:
  
            - ``'suffix tree'`` -- construct and use the suffix tree of the word
            - ``'naive'`` -- algorithm uses a sliding window
  
          OUTPUT:
  
          If ``n`` is an integer, returns the set of all distinct
          factors of length ``n``. If ``n`` is ``None``, returns the set
          of all distinct factors.
  
          EXAMPLES::
  
              sage: w = Word('121')
              sage: sorted(w.factor_set())
              [word: , word: 1, word: 12, word: 121, word: 2, word: 21]
              sage: sorted(w.factor_set(algorithm='naive'))
              [word: , word: 1, word: 12, word: 121, word: 2, word: 21]
  
          ::
  
              sage: w = Word('1213121')
              sage: for i in range(w.length()): sorted(w.factor_set(i))
              [word: ]
              [word: 1, word: 2, word: 3]
              [word: 12, word: 13, word: 21, word: 31]
              [word: 121, word: 131, word: 213, word: 312]
              [word: 1213, word: 1312, word: 2131, word: 3121]
              [word: 12131, word: 13121, word: 21312]
              [word: 121312, word: 213121]
  
          ::
  
              sage: w = Word([1,2,1,2,3])
              sage: s = w.factor_set()
              sage: sorted(s)
              [word: , word: 1, word: 12, word: 121, word: 1212, word: 12123, word: 123, word: 2, word: 21, word: 212, word: 2123, word: 23, word: 3]
  
          TESTS::
  
              sage: w = Word("xx")
              sage: s = w.factor_set()
              sage: sorted(s)
              [word: , word: x, word: xx]
  
          ::
  
              sage: Set(Word().factor_set())
              {word: }
  
          ::
  
              sage: w = Word(range(10), alphabet=range(10))
              sage: S1 = w.factor_set(3, algorithm='suffix tree')
              sage: S2 = w.factor_set(3, algorithm='naive')
              sage: S1 == S2
              True
          """
          if algorithm == 'suffix tree':
              return Set(self.factor_iterator(n))
          elif algorithm == 'naive':
              if n is None:
                  S = set([self[0:0]])
                  for n in range(1, self.length()+1):
                      for i in range(self.length()-n+1):
                          S.add(self[i:i+n])
                  return Set(S)
              else:
                  S = set()
                  for i in range(self.length()-n+1):
                      S.add(self[i:i+n])
                  return Set(S)
          else:
              raise ValueError('Unknown algorithm (={})'.format(algorithm))
  
      def topological_entropy(self, n):
          r"""
          Return the topological entropy for the factors of length ``n``.
  
          The topological entropy of a sequence `u` is defined as the
          exponential growth rate of the complexity of `u` as the length
          increases: `H_{top}(u)=\lim_{n\to\infty}\frac{\log_d(p_u(n))}{n}`
          where `d` denotes the cardinality of the alphabet and `p_u(n)` is
          the complexity function, i.e. the number of factors of length `n`
          in the sequence `u` [1].
  
          INPUT:
  
          - ``self`` -- a word defined over a finite alphabet
          -  ``n`` -- positive integer
  
          OUTPUT:
  
          real number (a symbolic expression)
  
          EXAMPLES::
  
              sage: W = Words([0, 1])
              sage: w = W([0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1])
              sage: t = w.topological_entropy(3); t
              1/3*log(7)/log(2)
              sage: n(t)
              0.935784974019201
  
          ::
  
              sage: w = words.ThueMorseWord()[:100]
              sage: topo = w.topological_entropy
              sage: for i in range(0, 41, 5):
              ....:     print("{} {}".format(i, n(topo(i), digits=5)))
              0 1.0000
              5 0.71699
              10 0.48074
              15 0.36396
              20 0.28774
              25 0.23628
              30 0.20075
              35 0.17270
              40 0.14827
  
          If no alphabet is specified, an error is raised::
  
              sage: w = Word(range(20))
              sage: w.topological_entropy(3)
              Traceback (most recent call last):
              ...
              TypeError: The word must be defined over a finite alphabet
  
          The following is ok::
  
              sage: W = Words(range(20))
              sage: w = W(range(20))
              sage: w.topological_entropy(3)
              1/3*log(18)/log(20)
  
          REFERENCES:
  
             [1] N. Pytheas Fogg, Substitutions in Dynamics, Arithmetics,
             and Combinatorics, Lecture Notes in Mathematics 1794, Springer
             Verlag. V. Berthe, S. Ferenczi, C. Mauduit and A. Siegel, Eds.
             (2002).
          """
          d = self.parent().alphabet().cardinality()
          if d is Infinity:
              raise TypeError("The word must be defined over a finite alphabet")
          if n == 0:
              return 1
          pn = self.number_of_factors(n)
          from sage.functions.all import log
          return log(pn, base=d)/n
  
      def rauzy_graph(self, n):
          r"""
          Return the Rauzy graph of the factors of length ``n`` of ``self``.
  
          The vertices are the factors of length `n` and there is an edge from
          `u` to `v` if `ua = bv` is a factor of length `n+1` for some letters
          `a` and `b`.
  
          INPUT:
  
          - ``n`` -- integer
  
          EXAMPLES::
  
              sage: w = Word(range(10)); w
              word: 0123456789
              sage: g = w.rauzy_graph(3); g
              Looped digraph on 8 vertices
              sage: WordOptions(identifier='')
              sage: g.vertices()
              [012, 123, 234, 345, 456, 567, 678, 789]
              sage: g.edges()
              [(012, 123, 3),
               (123, 234, 4),
               (234, 345, 5),
               (345, 456, 6),
               (456, 567, 7),
               (567, 678, 8),
               (678, 789, 9)]
              sage: WordOptions(identifier='word: ')
  
          ::
  
              sage: f = words.FibonacciWord()[:100]
              sage: f.rauzy_graph(8)
              Looped digraph on 9 vertices
  
          ::
  
              sage: w = Word('1111111')
              sage: g = w.rauzy_graph(3)
              sage: g.edges()
              [(word: 111, word: 111, word: 1)]
  
          ::
  
              sage: w = Word('111')
              sage: for i in range(5) : w.rauzy_graph(i)
              Looped multi-digraph on 1 vertex
              Looped digraph on 1 vertex
              Looped digraph on 1 vertex
              Looped digraph on 1 vertex
              Looped digraph on 0 vertices
  
          Multi-edges are allowed for the empty word::
  
              sage: W = Words('abcde')
              sage: w = W('abc')
              sage: w.rauzy_graph(0)
              Looped multi-digraph on 1 vertex
              sage: _.edges()
              [(word: , word: , word: a),
               (word: , word: , word: b),
               (word: , word: , word: c)]
          """
          from sage.graphs.digraph import DiGraph
          multiedges = n == 0
          g = DiGraph(loops=True, multiedges=multiedges)
          if n == self.length():
              g.add_vertex(self)
          else:
              for w in self.factor_iterator(n+1):
                  u = w[:-1]
                  v = w[1:]
                  a = w[-1:]
                  g.add_edge(u,v,a)
          return g
  
      def reduced_rauzy_graph(self, n):
          r"""
          Return the reduced Rauzy graph of order ``n`` of ``self``.
  
          INPUT:
  
          - ``n`` -- a non-negative integer. Every vertex of a reduced
            Rauzy graph of order ``n`` is a factor of length ``n`` of ``self``.
  
          OUTPUT:
  
          a looped multi-digraph
  
          DEFINITION:
  
          For infinite periodic words (resp. for finite words of type `u^i
          u[0:j]`), the reduced Rauzy graph of order `n` (resp. for `n`
          smaller or equal to `(i-1)|u|+j`) is the directed graph whose
          unique vertex is the prefix `p` of length `n` of ``self`` and which has
          an only edge which is a loop on `p` labelled by `w[n+1:|w|] p`
          where `w` is the unique return word to `p`.
  
          In other cases, it is the directed graph defined as followed.  Let
          `G_n` be the Rauzy graph of order `n` of self. The vertices are the
          vertices of `G_n` that are either special or not prolongable to the
          right or to the left. For each couple (`u`, `v`) of such vertices
          and each directed path in `G_n` from `u` to `v` that contains no
          other vertices that are special, there is an edge from `u` to `v`
          in the reduced Rauzy graph of order `n` whose label is the label of
          the path in `G_n`.
  
          .. NOTE::
  
              In the case of infinite recurrent non-periodic words, this
              definition corresponds to the following one that can be found in
              [1] and [2]  where a simple path is a path that begins with a
              special factor, ends with a special factor and contains no
              other vertices that are special:
  
              The reduced Rauzy graph of factors of length `n` is obtained
              from `G_n` by replacing each simple path `P=v_1 v_2 ...
              v_{\ell}` with an edge `v_1 v_{\ell}` whose label is the
              concatenation of the labels of the edges of `P`.
  
          EXAMPLES::
  
              sage: w = Word(range(10)); w
              word: 0123456789
              sage: g = w.reduced_rauzy_graph(3); g
              Looped multi-digraph on 2 vertices
              sage: g.vertices()
              [word: 012, word: 789]
              sage: g.edges()
              [(word: 012, word: 789, word: 3456789)]
  
          For the Fibonacci word::
  
              sage: f = words.FibonacciWord()[:100]
              sage: g = f.reduced_rauzy_graph(8);g
              Looped multi-digraph on 2 vertices
              sage: g.vertices()
              [word: 01001010, word: 01010010]
              sage: g.edges()
              [(word: 01001010, word: 01010010, word: 010), (word: 01010010, word: 01001010, word: 01010), (word: 01010010, word: 01001010, word: 10)]
  
          For periodic words::
  
              sage: from itertools import cycle
              sage: w = Word(cycle('abcd'))[:100]
              sage: g = w.reduced_rauzy_graph(3)
              sage: g.edges()
              [(word: abc, word: abc, word: dabc)]
  
          ::
  
              sage: w = Word('111')
              sage: for i in range(5) : w.reduced_rauzy_graph(i)
              Looped digraph on 1 vertex
              Looped digraph on 1 vertex
              Looped digraph on 1 vertex
              Looped multi-digraph on 1 vertex
              Looped multi-digraph on 0 vertices
  
          For ultimately periodic words::
  
              sage: sigma = WordMorphism('a->abcd,b->cd,c->cd,d->cd')
              sage: w = sigma.fixed_point('a')[:100]; w
              word: abcdcdcdcdcdcdcdcdcdcdcdcdcdcdcdcdcdcdcd...
              sage: g = w.reduced_rauzy_graph(5)
              sage: g.vertices()
              [word: abcdc, word: cdcdc]
              sage: g.edges()
              [(word: abcdc, word: cdcdc, word: dc), (word: cdcdc, word: cdcdc, word: dc)]
  
          AUTHOR:
  
          Julien Leroy (March 2010): initial version
  
          REFERENCES:
  
          - [1] M. Bucci et al.  A. De Luca, A. Glen, L. Q. Zamboni, A
            connection between palindromic and factor complexity using
            return words," Advances in Applied Mathematics 42 (2009) 60-74.
  
          - [2] L'ubomira Balkova, Edita Pelantova, and Wolfgang Steiner.
            Sequences with constant number of return words. Monatsh. Math,
            155 (2008) 251-263.
          """
          from sage.graphs.digraph import DiGraph
          from copy import copy
          g = copy(self.rauzy_graph(n))
          # Otherwise it changes the rauzy_graph function.
          l = [v for v in g if g.in_degree(v)==1 and g.out_degree(v)==1]
          if g.num_verts() != 0 and len(l) == g.num_verts():
              # In this case, the Rauzy graph is simply a cycle.
              g = DiGraph()
              g.allow_loops(True)
              g.add_vertex(self[:n])
              g.add_edge(self[:n],self[:n],self[n:n+len(l)])
          else:
              g.allow_loops(True)
              g.allow_multiple_edges(True)
              for v in l:
                  [i] = g.neighbors_in(v)
                  [o] = g.neighbors_out(v)
                  g.add_edge(i,o,g.edge_label(i,v)[0]*g.edge_label(v,o)[0])
                  g.delete_vertex(v)
          return g
  
      def left_special_factors_iterator(self, n=None):
          r"""
          Return an iterator over the left special factors (of length ``n``).
  
          A factor `u` of a word `w` is *left special* if there are
          two distinct letters `a` and `b` such that `au` and `bu`
          are factors of `w`.
  
          INPUT:
  
          - ``n`` -- integer (optional, default: ``None``). If ``None``, it returns
            an iterator over all left special factors.
  
          EXAMPLES::
  
              sage: alpha, beta, x = 0.54, 0.294, 0.1415
              sage: w = words.CodingOfRotationWord(alpha, beta, x)[:40]
              sage: sorted(w.left_special_factors_iterator(3))
              [word: 000, word: 010]
              sage: sorted(w.left_special_factors_iterator(4))
              [word: 0000, word: 0101]
              sage: sorted(w.left_special_factors_iterator(5))
              [word: 00000, word: 01010]
          """
          if n is None:
              for i in range(self.length()):
                  for w in self.left_special_factors_iterator(i):
                      yield w
          else:
              left_extensions = defaultdict(set)
              for w in self.factor_iterator(n+1):
                  v = w[1:]
                  left_extensions[v].add(w[0])
              for v in left_extensions:
                  if len(left_extensions[v]) > 1:
                      yield v
  
      def left_special_factors(self, n=None):
          r"""
          Return the left special factors (of length ``n``).
  
          A factor `u` of a word `w` is *left special* if there are
          two distinct letters `a` and `b` such that `au` and `bu`
          are factors of `w`.
  
          INPUT:
  
          - ``n`` -- integer (optional, default: ``None``). If ``None``, it
            returns all left special factors.
  
          OUTPUT:
  
          a list of words
  
          EXAMPLES::
  
              sage: alpha, beta, x = 0.54, 0.294, 0.1415
              sage: w = words.CodingOfRotationWord(alpha, beta, x)[:40]
              sage: for i in range(5):
              ....:     print("{} {}".format(i, sorted(w.left_special_factors(i))))
              0 [word: ]
              1 [word: 0]
              2 [word: 00, word: 01]
              3 [word: 000, word: 010]
              4 [word: 0000, word: 0101]
          """
          return list(self.left_special_factors_iterator(n))
  
      def right_special_factors_iterator(self, n=None):
          r"""
          Return an iterator over the right special factors (of length ``n``).
  
          A factor `u` of a word `w` is *right special* if there are
          two distinct letters `a` and `b` such that `ua` and `ub`
          are factors of `w`.
  
          INPUT:
  
          - ``n`` -- integer (optional, default: ``None``). If ``None``, it returns
            an iterator over all right special factors.
  
          EXAMPLES::
  
              sage: alpha, beta, x = 0.61, 0.54, 0.3
              sage: w = words.CodingOfRotationWord(alpha, beta, x)[:40]
              sage: sorted(w.right_special_factors_iterator(3))
              [word: 010, word: 101]
              sage: sorted(w.right_special_factors_iterator(4))
              [word: 0101, word: 1010]
              sage: sorted(w.right_special_factors_iterator(5))
              [word: 00101, word: 11010]
          """
          if n is None:
              for i in range(self.length()):
                  for w in self.right_special_factors_iterator(i):
                      yield w
          else:
              right_extensions = defaultdict(set)
              for w in self.factor_iterator(n+1):
                  v = w[:-1]
                  right_extensions[v].add(w[-1])
              for v in right_extensions:
                  if len(right_extensions[v]) > 1:
                      yield v
  
      def right_special_factors(self, n=None):
          r"""
          Return the right special factors (of length ``n``).
  
          A factor `u` of a word `w` is *right special* if there are
          two distinct letters `a` and `b` such that `ua` and `ub`
          are factors of `w`.
  
          INPUT:
  
          - ``n`` -- integer (optional, default: ``None``). If ``None``, it returns
            all right special factors.
  
          OUTPUT:
  
          a list of words
  
          EXAMPLES::
  
              sage: w = words.ThueMorseWord()[:30]
              sage: for i in range(5):
              ....:     print("{} {}".format(i, sorted(w.right_special_factors(i))))
              0 [word: ]
              1 [word: 0, word: 1]
              2 [word: 01, word: 10]
              3 [word: 001, word: 010, word: 101, word: 110]
              4 [word: 0110, word: 1001]
          """
          return list(self.right_special_factors_iterator(n))
  
      def bispecial_factors_iterator(self, n=None):
          r"""
          Return an iterator over the bispecial factors (of length ``n``).
  
          A factor `u` of a word `w` is *bispecial* if it is right special
          and left special.
  
          INPUT:
  
          - ``n`` -- integer (optional, default: ``None``). If ``None``, it returns
            an iterator over all bispecial factors.
  
          EXAMPLES::
  
              sage: w = words.ThueMorseWord()[:30]
              sage: for i in range(10):
              ....:     for u in sorted(w.bispecial_factors_iterator(i)):
              ....:         print("{} {}".format(i,u))
              0
              1 0
              1 1
              2 01
              2 10
              3 010
              3 101
              4 0110
              4 1001
              6 011001
              6 100110
              8 10010110
  
          ::
  
              sage: key = lambda u : (len(u), u)
              sage: for u in sorted(w.bispecial_factors_iterator(), key=key): u
              word:
              word: 0
              word: 1
              word: 01
              word: 10
              word: 010
              word: 101
              word: 0110
              word: 1001
              word: 011001
              word: 100110
              word: 10010110
          """
          if n is None:
              for i in range(self.length()):
                  for w in self.bispecial_factors_iterator(i):
                      yield w
          else:
              left_extensions = defaultdict(set)
              right_extensions = defaultdict(set)
              for w in self.factor_iterator(n+2):
                  v = w[1:-1]
                  left_extensions[v].add(w[0])
                  right_extensions[v].add(w[-1])
              for v in left_extensions:
                  if (len(left_extensions[v]) > 1 and
                      len(right_extensions[v]) > 1):
                      yield v
  
      def bispecial_factors(self, n=None):
          r"""
          Return the bispecial factors (of length ``n``).
  
          A factor `u` of a word `w` is *bispecial* if it is right special
          and left special.
  
          INPUT:
  
          - ``n`` -- integer (optional, default: ``None``). If ``None``, it returns
            all bispecial factors.
  
          OUTPUT:
  
          a list of words
  
          EXAMPLES::
  
              sage: w = words.FibonacciWord()[:30]
              sage: w.bispecial_factors()
              [word: , word: 0, word: 010, word: 010010, word: 01001010010]
  
          ::
  
              sage: w = words.ThueMorseWord()[:30]
              sage: for i in range(10):
              ....:     print("{} {}".format(i, sorted(w.bispecial_factors(i))))
              0 [word: ]
              1 [word: 0, word: 1]
              2 [word: 01, word: 10]
              3 [word: 010, word: 101]
              4 [word: 0110, word: 1001]
              5 []
              6 [word: 011001, word: 100110]
              7 []
              8 [word: 10010110]
              9 []
          """
          return list(self.bispecial_factors_iterator(n))
  
      def number_of_left_special_factors(self, n):
          r"""
          Return the number of left special factors of length ``n``.
  
          A factor `u` of a word `w` is *left special* if there are
          two distinct letters `a` and `b` such that `au` and `bu`
          are factors of `w`.
  
          INPUT:
  
          - ``n`` -- integer
  
          OUTPUT:
  
          a non-negative integer
  
          EXAMPLES::
  
              sage: w = words.FibonacciWord()[:100]
              sage: [w.number_of_left_special_factors(i) for i in range(10)]
              [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
  
          ::
  
              sage: w = words.ThueMorseWord()[:100]
              sage: [w.number_of_left_special_factors(i) for i in range(10)]
              [1, 2, 2, 4, 2, 4, 4, 2, 2, 4]
          """
          it = self.left_special_factors_iterator(n)
          return sum(1 for _ in it)
  
      def number_of_right_special_factors(self, n):
          r"""
          Return the number of right special factors of length ``n``.
  
          A factor `u` of a word `w` is *right special* if there are
          two distinct letters `a` and `b` such that `ua` and `ub`
          are factors of `w`.
  
          INPUT:
  
          - ``n`` -- integer
  
          OUTPUT:
  
          a non-negative integer
  
          EXAMPLES::
  
              sage: w = words.FibonacciWord()[:100]
              sage: [w.number_of_right_special_factors(i) for i in range(10)]
              [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
  
          ::
  
              sage: w = words.ThueMorseWord()[:100]
              sage: [w.number_of_right_special_factors(i) for i in range(10)]
              [1, 2, 2, 4, 2, 4, 4, 2, 2, 4]
          """
          it = self.right_special_factors_iterator(n)
          return sum(1 for _ in it)
  
      def commutes_with(self, other):
          r"""
          Return ``True`` if ``self`` commutes with ``other``, and ``False`` otherwise.
  
          EXAMPLES::
  
              sage: Word('12').commutes_with(Word('12'))
              True
              sage: Word('12').commutes_with(Word('11'))
              False
              sage: Word().commutes_with(Word('21'))
              True
          """
          return (self * other) == (other * self)
  
      def conjugate(self, pos):
          r"""
          Return the conjugate at ``pos`` of ``self``.
  
          ``pos`` can be any integer, the distance used is the modulo by the length
          of ``self``.
  
          EXAMPLES::
  
              sage: Word('12112').conjugate(1)
              word: 21121
              sage: Word().conjugate(2)
              word:
              sage: Word('12112').conjugate(8)
              word: 12121
              sage: Word('12112').conjugate(-1)
              word: 21211
          """
          if self.is_empty():
              return self
          pos_mod = pos % self.length()
          return self[pos_mod:] * self[:pos_mod]
  
      def _conjugates_list(self):
          r"""
          Return the list of conjugates of ``self``, ordered from the `0`-th to the
          `(L-1)`-st conjugate, where `L` is the length of ``self``.
  
          TESTS::
  
              sage: Word('cbbca')._conjugates_list()
              [word: cbbca, word: bbcac, word: bcacb, word: cacbb, word: acbbc]
              sage: Word('abcabc')._conjugates_list()
              [word: abcabc,
               word: bcabca,
               word: cabcab,
               word: abcabc,
               word: bcabca,
               word: cabcab]
              sage: Word()._conjugates_list()
              [word: ]
              sage: Word('a')._conjugates_list()
              [word: a]
          """
          S = [self]
          for i in range(1,self.length()):
              S.append(self.conjugate(i))
          return S
  
      def conjugates_iterator(self):
          r"""
          Return an iterator over the conjugates of ``self``.
  
          EXAMPLES::
  
              sage: it = Word(range(4)).conjugates_iterator()
              sage: for w in it: w
              word: 0123
              word: 1230
              word: 2301
              word: 3012
          """
          yield self
          for i in range(1, self.primitive_length()):
              yield self.conjugate(i)
  
      def conjugates(self):
          r"""
          Return the list of unique conjugates of ``self``.
  
          EXAMPLES::
  
              sage: Word(range(6)).conjugates()
              [word: 012345,
               word: 123450,
               word: 234501,
               word: 345012,
               word: 450123,
               word: 501234]
              sage: Word('cbbca').conjugates()
              [word: cbbca, word: bbcac, word: bcacb, word: cacbb, word: acbbc]
  
          The result contains each conjugate only once::
  
              sage: Word('abcabc').conjugates()
              [word: abcabc, word: bcabca, word: cabcab]
  
          TESTS::
  
              sage: Word().conjugates()
              [word: ]
              sage: Word('a').conjugates()
              [word: a]
          """
          return list(self.conjugates_iterator())
  
      def conjugate_position(self, other):
          r"""
          Return the position where ``self`` is conjugate with ``other``.
          Return ``None`` if there is no such position.
  
          EXAMPLES::
  
              sage: Word('12113').conjugate_position(Word('31211'))
              1
              sage: Word('12131').conjugate_position(Word('12113')) is None
              True
              sage: Word().conjugate_position(Word('123')) is None
              True
  
          TESTS:
  
          We check that :trac:`11128` is fixed::
  
              sage: w = Word([0,0,1,0,2,1])
              sage: [w.conjugate(i).conjugate_position(w) for i in range(w.length())]
              [0, 1, 2, 3, 4, 5]
          """
          if self.length() != other.length():
              return None
          other_square = other * other
          pos = other_square.find(self)
          return pos if pos != -1 else None
  
      def is_conjugate_with(self, other):
          r"""
          Return ``True`  if ``self`` is a conjugate of ``other``, and ``False`` otherwise.
  
          INPUT:
  
          - ``other`` -- a finite word
  
          OUTPUT:
  
          bool
  
          EXAMPLES::
  
              sage: w = Word([0..20])
              sage: z = Word([7..20] + [0..6])
              sage: w
              word: 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20
              sage: z
              word: 7,8,9,10,11,12,13,14,15,16,17,18,19,20,0,1,2,3,4,5,6
              sage: w.is_conjugate_with(z)
              True
              sage: z.is_conjugate_with(w)
              True
              sage: u = Word([4]*21)
              sage: u.is_conjugate_with(w)
              False
              sage: u.is_conjugate_with(z)
              False
  
          Both words must be finite::
  
              sage: w = Word(iter([2]*100),length='unknown')
              sage: z = Word([2]*100)
              sage: z.is_conjugate_with(w) #TODO: Not implemented for word of unknown length
              True
              sage: wf = Word(iter([2]*100),length='finite')
              sage: z.is_conjugate_with(wf)
              True
              sage: wf.is_conjugate_with(z)
              True
  
          TESTS::
  
              sage: Word('11213').is_conjugate_with(Word('31121'))
              True
              sage: Word().is_conjugate_with(Word('123'))
              False
              sage: Word('112131').is_conjugate_with(Word('11213'))
              False
              sage: Word('12131').is_conjugate_with(Word('11213'))
              True
  
          We make sure that :trac:`11128` is fixed::
  
              sage: Word('abaa').is_conjugate_with(Word('aaba'))
              True
              sage: Word('aaba').is_conjugate_with(Word('abaa'))
              True
          """
          return self.length() == other.length() and self.is_factor(other * other)
  
      def is_cadence(self, seq):
          r"""
          Return ``True`` if ``seq`` is a cadence of ``self``, and ``False`` otherwise.
  
          A *cadence* is an increasing sequence of indexes that all map to
          the same letter.
  
          EXAMPLES::
  
              sage: Word('121132123').is_cadence([0, 2, 6])
              True
              sage: Word('121132123').is_cadence([0, 1, 2])
              False
              sage: Word('121132123').is_cadence([])
              True
          """
          if len(seq) == 0:
              return True
          try:
              it = iter(self)
              s = next(islice(it, seq[0], None))
              for i in range(1, len(seq)):
                  steps = seq[i] - seq[i-1]
                  for n in range(steps-1): next(it)
                  if next(it) != s:
                      return False
          except StopIteration:
              return False
          return True
  
      def longest_forward_extension(self, x, y):
          r"""
          Compute the length of the longest factor of ``self`` that
          starts at ``x`` and that matches a factor that starts at ``y``.
  
          INPUT:
  
          - ``x``, ``y`` -- positions in ``self``
  
          EXAMPLES::
  
              sage: w = Word('0011001')
              sage: w.longest_forward_extension(0, 4)
              3
              sage: w.longest_forward_extension(0, 2)
              0
  
          The method also accepts negative positions indicating the distance from
          the end of the word (in order to be consist with how negative indices
          work with lists). For instance, for a word of length `7`, using
          positions `-3` and `2` is the same as using positions `4` and `2`::
  
              sage: w.longest_forward_extension(1, -2)
              2
              sage: w.longest_forward_extension(4, -3)
              3
  
          TESTS::
  
              sage: w = Word('0011001')
              sage: w.longest_forward_extension(-10, 2)
              Traceback (most recent call last):
              ...
              ValueError: x and y must be valid positions in self
  
          """
          length = self.length()
          if not (-length <= x < length and -length <= y < length):
              raise ValueError("x and y must be valid positions in self")
          if x < 0:
              x = x + length
          if y < 0:
              y = y + length
          l = 0
          while x < length and y < length and self[x] == self[y]:
              l += 1
              x += 1
              y += 1
          return l
  
      def longest_backward_extension(self, x, y):
          r"""
          Compute the length of the longest factor of ``self`` that
          ends at ``x`` and that matches a factor that ends at ``y``.
  
          INPUT:
  
          - ``x``, ``y`` -- positions in ``self``
  
          EXAMPLES::
  
              sage: w = Word('0011001')
              sage: w.longest_backward_extension(6, 2)
              3
              sage: w.longest_backward_extension(1, 4)
              1
              sage: w.longest_backward_extension(1, 3)
              0
  
          The method also accepts negative positions indicating the distance from
          the end of the word (in order to be consist with how negative indices
          work with lists). For instance, for a word of length `7`, using
          positions `6` and `-5` is the same as using positions `6` and `2`::
  
              sage: w.longest_backward_extension(6, -5)
              3
              sage: w.longest_backward_extension(-6, 4)
              1
  
          TESTS::
  
              sage: w = Word('0011001')
              sage: w.longest_backward_extension(4, 23)
              Traceback (most recent call last):
              ...
              ValueError: x and y must be valid positions in self
              sage: w.longest_backward_extension(-9, 4)
              Traceback (most recent call last):
              ...
              ValueError: x and y must be valid positions in self
          """
          length = self.length()
          if not (-length <= x < length and -length <= y < length):
              raise ValueError("x and y must be valid positions in self")
          if x < 0:
              x = x + length
          if y < 0:
              y = y + length
          l = 0
          while x >= 0 and y >= 0 and self[x] == self[y]:
              l += 1
              x -= 1
              y -= 1
          return l
  
      def longest_common_suffix(self, other):
          r"""
          Return the longest common suffix of ``self`` and ``other``.
  
          EXAMPLES::
  
              sage: w = Word('112345678')
              sage: u = Word('1115678')
              sage: w.longest_common_suffix(u)
              word: 5678
              sage: u.longest_common_suffix(u)
              word: 1115678
              sage: u.longest_common_suffix(w)
              word: 5678
              sage: w.longest_common_suffix(w)
              word: 112345678
              sage: y = Word('549332345')
              sage: w.longest_common_suffix(y)
              word:
  
          TESTS:
  
          With the empty word::
  
              sage: w.longest_common_suffix(Word())
              word:
              sage: Word().longest_common_suffix(w)
              word:
              sage: Word().longest_common_suffix(Word())
              word:
  
          With an infinite word::
  
              sage: t=words.ThueMorseWord('ab')
              sage: w.longest_common_suffix(t)
              Traceback (most recent call last):
              ...
              TypeError: other must be a finite word
          """
          if not isinstance(other, FiniteWord_class):
              raise TypeError("other must be a finite word")
  
          if self.is_empty():
              return self
          if other.is_empty():
              return other
  
          iter = enumerate(zip(reversed(self), reversed(other)))
          i,(b,c) = next(iter)
          if b != c:
              #In this case, return the empty word
              return self[:0]
  
          for i,(b,c) in iter:
              if b != c:
                  return self[-i:]
          else:
              return self[-i-1:]
  
      def is_palindrome(self, f=None):
          r"""
          Return ``True`` if ``self`` is a palindrome (or a ``f``-palindrome), and
          ``False`` otherwise.
  
          Let `f : \Sigma \rightarrow \Sigma` be an involution that extends
          to a morphism on `\Sigma^*`. We say that `w\in\Sigma^*` is a
          *`f`-palindrome* if `w=f(\tilde{w})` [1]. Also called
          *`f`-pseudo-palindrome* [2].
  
          INPUT:
  
          - ``f`` -- involution (default: ``None``) on the alphabet of ``self``.
            It must be callable on letters as well as words (e.g.
            :class:`~sage.combinat.words.morphism.WordMorphism`). The
            default value corresponds to usual palindromes, i.e., ``f``
            equal to the identity.
  
          EXAMPLES::
  
              sage: Word('esope reste ici et se repose').is_palindrome()
              False
              sage: Word('esoperesteicietserepose').is_palindrome()
              True
              sage: Word('I saw I was I').is_palindrome()
              True
              sage: Word('abbcbba').is_palindrome()
              True
              sage: Word('abcbdba').is_palindrome()
              False
  
          Some `f`-palindromes::
  
              sage: f = WordMorphism('a->b,b->a')
              sage: Word('aababb').is_palindrome(f)
              True
  
          ::
  
              sage: f = WordMorphism('a->b,b->a,c->c')
              sage: Word('abacbacbab').is_palindrome(f)
              True
  
          ::
  
              sage: f = WordMorphism({'a':'b','b':'a'})
              sage: Word('aababb').is_palindrome(f)
              True
  
          ::
  
              sage: f = WordMorphism({0:[1],1:[0]})
              sage: w = words.ThueMorseWord()[:8]; w
              word: 01101001
              sage: w.is_palindrome(f)
              True
  
          The word must be in the domain of the involution::
  
              sage: f = WordMorphism('a->a')
              sage: Word('aababb').is_palindrome(f)
              Traceback (most recent call last):
              ...
              KeyError: 'b'
  
          TESTS:
  
          If the given involution is not an involution::
  
              sage: f = WordMorphism('a->b,b->b')
              sage: Word('abab').is_palindrome(f)
              Traceback (most recent call last):
              ...
              TypeError: self (=a->b, b->b) is not an endomorphism
  
          ::
  
              sage: Y = Word
              sage: Y().is_palindrome()
              True
              sage: Y('a').is_palindrome()
              True
              sage: Y('ab').is_palindrome()
              False
              sage: Y('aba').is_palindrome()
              True
              sage: Y('aa').is_palindrome()
              True
              sage: E = WordMorphism('a->b,b->a')
              sage: Y().is_palindrome(E)
              True
              sage: Y('a').is_palindrome(E)
              False
              sage: Y('ab').is_palindrome(E)
              True
              sage: Y('aa').is_palindrome(E)
              False
              sage: Y('aba').is_palindrome(E)
              False
              sage: Y('abab').is_palindrome(E)
              True
  
          REFERENCES:
  
          -   [1] S. Labbé, Propriétés combinatoires des `f`-palindromes,
              Mémoire de maîtrise en Mathématiques, Montréal, UQAM, 2008,
              109 pages.
          -   [2] V. Anne, L.Q. Zamboni, I. Zorca, Palindromes and Pseudo-
              Palindromes in Episturmian and Pseudo-Palindromic Infinite Words,
              in : S. Brlek, C. Reutenauer (Eds.), Words 2005, Publications du
              LaCIM, Vol. 36 (2005) 91--100.
          """
          l = self.length()
          if f is None:
              return self[:l//2] == self[l//2 + l%2:].reversal()
          else:
              from sage.combinat.words.morphism import WordMorphism
              if not isinstance(f, WordMorphism):
                  f = WordMorphism(f)
              if not f.is_involution():
                  raise ValueError("f must be an involution")
              return self[:l//2 + l%2] == f(self[l//2:].reversal())
  
      def lps(self, f=None, l=None):
          r"""
          Return the longest palindromic (or ``f``-palindromic) suffix of ``self``.
  
          INPUT:
  
          - ``f`` -- involution (default: ``None``) on the alphabet of ``self``.
            It must be callable on letters as well as words (e.g. ``WordMorphism``).
          - ``l`` -- integer (default: ``None``) the length of the longest
            palindrome suffix of ````self[:-1]````, if known.
  
          OUTPUT:
  
          word -- If ``f`` is ``None``, the longest palindromic suffix of ``self``;
          otherwise, the longest ``f``-palindromic suffix of ``self``.
  
          EXAMPLES::
  
              sage: Word('0111').lps()
              word: 111
              sage: Word('011101').lps()
              word: 101
              sage: Word('6667').lps()
              word: 7
              sage: Word('abbabaab').lps()
              word: baab
              sage: Word().lps()
              word:
              sage: f = WordMorphism('a->b,b->a')
              sage: Word('abbabaab').lps(f=f)
              word: abbabaab
              sage: w = Word('33412321')
              sage: w.lps(l=3)
              word: 12321
              sage: Y = Word
              sage: w = Y('01101001')
              sage: w.lps(l=2)
              word: 1001
              sage: w.lps()
              word: 1001
              sage: w.lps(l=None)
              word: 1001
              sage: Y().lps(l=2)
              Traceback (most recent call last):
              ...
              IndexError: list index out of range
              sage: v = Word('abbabaab')
              sage: pal = v[:0]
              sage: for i in range(1, v.length()+1):
              ....:   pal = v[:i].lps(l=pal.length())
              ....:   pal
              word: a
              word: b
              word: bb
              word: abba
              word: bab
              word: aba
              word: aa
              word: baab
              sage: f = WordMorphism('a->b,b->a')
              sage: v = Word('abbabaab')
              sage: pal = v[:0]
              sage: for i in range(1, v.length()+1):
              ....:   pal = v[:i].lps(f=f, l=pal.length())
              ....:   pal
              word:
              word: ab
              word:
              word: ba
              word: ab
              word: baba
              word: bbabaa
              word: abbabaab
          """
          #If the length of the lps of self[:-1] is not known:
          if l is None:
              l = self.lps_lengths(f)[-1]
              return self[len(self)-l:]
  
          #If l == w[:-1].length(), there is no shortcut
          if self.length() == l + 1:
              return self.lps(f=f)
  
          #Obtain the letter to the left (g) and to the right (d) of the
          #precedent lps of self
          g = self[-l-2]
          d = self[-1]
  
          #If the word g*d is a `f`-palindrome, the result follows
          if f is None:
              if g == d:
                  return self[-l-2:]
              else:
                  #Otherwise, the length of the lps of self is smallest than l+2
                  return self[-l-1:].lps()
          else:
              from sage.combinat.words.morphism import WordMorphism
              f = WordMorphism(f)
              if f(g)[0] == d:
                  return self[-l-2:]
              else:
                  return self[-l-1:].lps(f=f)
  
      @cached_method
      def palindromic_lacunas_study(self, f=None):
          r"""
          Return interesting statistics about longest (``f``-)palindromic suffixes
          and lacunas of ``self`` (see [1] and [2]).
  
          Note that a word `w` has at most `|w| + 1` different palindromic factors
          (see [3]). For `f`-palindromes (or pseudopalidromes or theta-palindromes),
          the maximum number of `f`-palindromic factors is `|w|+1-g_f(w)`, where
          `g_f(w)` is the number of pairs `\{a, f(a)\}` such that `a` is a letter,
          `a` is not equal to `f(a)`, and `a` or `f(a)` occurs in `w`, see [4].
  
          INPUT:
  
          - ``f`` -- involution (default: ``None``) on the alphabet of ``self``.
            It must be callable on letters as well as words (e.g. ``WordMorphism``).
            The default value corresponds to usual palindromes, i.e.,
            ``f`` equal to the identity.
  
          OUTPUT:
  
          - ``list`` -- list of the length of the longest palindromic
            suffix (lps) for each non-empty prefix of ``self``
          - ``list`` -- list of all the lacunas, i.e. positions where there is no
            unioccurrent lps
          - ``set`` -- set of palindromic factors of ``self``
  
          EXAMPLES::
  
              sage: a,b,c = Word('abbabaabbaab').palindromic_lacunas_study()
              sage: a
              [1, 1, 2, 4, 3, 3, 2, 4, 2, 4, 6, 8]
              sage: b
              [8, 9]
              sage: c          # random order
              set([word: , word: b, word: bab, word: abba, word: bb, word: aa, word: baabbaab, word: baab, word: aba, word: aabbaa, word: a])
  
          ::
  
              sage: f = WordMorphism('a->b,b->a')
              sage: a,b,c = Word('abbabaab').palindromic_lacunas_study(f=f)
              sage: a
              [0, 2, 0, 2, 2, 4, 6, 8]
              sage: b
              [0, 2, 4]
              sage: c           # random order
              set([word: , word: ba, word: baba, word: ab, word: bbabaa, word: abbabaab])
              sage: c == set([Word(), Word('ba'), Word('baba'), Word('ab'), Word('bbabaa'), Word('abbabaab')])
              True
  
          REFERENCES:
  
          -   [1] A. Blondin-Massé, S. Brlek, S. Labbé, Palindromic lacunas
              of the Thue-Morse word, Proc. GASCOM 2008 (June 16-20 2008,
              Bibbiena, Arezzo-Italia), 53--67.
          -   [2] A. Blondin-Massé, S. Brlek, A. Frosini, S. Labbé, S. Rinaldi,
              Reconstructing words from a fixed palindromic length sequence,
              Proc. TCS 2008, 5th IFIP International Conference on Theoretical
              Computer Science (September 8-10 2008, Milano, Italia), accepted.
          -   [3] X. Droubay, J. Justin, G. Pirillo, Episturmian words and
              some constructions of de Luca and Rauzy, Theoret. Comput. Sci.
              255 (2001) 539--553.
          -   [4] Š. Starosta, On Theta-palindromic Richness, Theoret. Comp.
              Sci. 412 (2011) 1111--1121
          """
          #Initialize the results of computations
          palindromes = set()
          lengths_lps = [None] * self.length()
          lacunas = []
  
          #Initialize the first lps
          pal = self[:0]
          palindromes.add(pal)
  
          #For all the non-empty prefixes of self,
          for i in range(self.length()):
  
              #Compute its longest `f`-palindromic suffix using the preceding lps (pal)
              pal = self[:i+1].lps(l=pal.length(),f=f)
  
              lengths_lps[i] = pal.length()
  
              if pal in palindromes:
                  lacunas.append(i)
              else :
                  palindromes.add(pal)
  
          return lengths_lps, lacunas, palindromes
  
      def lengths_lps(self, f=None):
          r"""
          Return the list of the length of the longest palindromic
          suffix (lps) for each non-empty prefix of ``self``.
  
          It corresponds to the function `G_w` defined in [1].
  
          INPUT:
  
          - ``f`` -- involution (default: ``None``) on the alphabet of ``self``. It must
            be callable on letters as well as words (e.g. ``WordMorphism``).
  
          OUTPUT:
  
          a list -- list of the length of the longest palindromic
          suffix (lps) for each non-empty prefix of ``self``
  
          EXAMPLES::
  
              sage: Word().lengths_lps()
              []
              sage: Word('a').lengths_lps()
              [1]
              sage: Word('aaa').lengths_lps()
              [1, 2, 3]
              sage: Word('abbabaabbaab').lengths_lps()
              [1, 1, 2, 4, 3, 3, 2, 4, 2, 4, 6, 8]
  
          ::
  
              sage: f = WordMorphism('a->b,b->a')
              sage: Word('abbabaabbaab').lengths_lps(f)
              [0, 2, 0, 2, 2, 4, 6, 8, 4, 6, 4, 6]
  
          ::
  
              sage: f = WordMorphism({5:[8],8:[5]})
              sage: Word([5,8,5,5,8,8,5,5,8,8,5,8,5]).lengths_lps(f)
              [0, 2, 2, 0, 2, 4, 6, 4, 6, 8, 10, 12, 4]
  
          REFERENCES:
  
          -   [1] A. Blondin-Massé, S. Brlek, A. Frosini, S. Labbé,
              S. Rinaldi, Reconstructing words from a fixed palindromic length
              sequence, Proc. TCS 2008, 5th IFIP International Conference on
              Theoretical Computer Science (September 8-10 2008, Milano,
              Italia), accepted.
          """
          return self.palindromic_lacunas_study(f=f)[0]
  
      def lacunas(self, f=None):
          r"""
          Return the list of all the lacunas of ``self``.
  
          A *lacuna* is a position in a word where the longest (`f`-)palindromic
          suffix is not unioccurrent (see [1]).
  
          INPUT:
  
          - ``f`` -- involution (default: ``None``) on the alphabet of ``self``. It must
            be callable on letters as well as words (e.g. ``WordMorphism``). The
            default value corresponds to usual palindromes, i.e., ``f`` equal to
            the identity.
  
          OUTPUT:
  
          a list -- list of all the lacunas of self
  
          EXAMPLES::
  
              sage: w = Word([0,1,1,2,3,4,5,1,13,3])
              sage: w.lacunas()
              [7, 9]
              sage: words.ThueMorseWord()[:100].lacunas()
              [8, 9, 24, 25, 32, 33, 34, 35, 36, 37, 38, 39, 96, 97, 98, 99]
              sage: f = WordMorphism({0:[1],1:[0]})
              sage: words.ThueMorseWord()[:50].lacunas(f)
              [0, 2, 4, 12, 16, 17, 18, 19, 48, 49]
  
          REFERENCES:
  
          -   [1] A. Blondin-Massé, S. Brlek, S. Labbé, Palindromic lacunas
              of the Thue-Morse word, Proc. GASCOM 2008 (June 16-20 2008,
              Bibbiena, Arezzo-Italia), 53--67.
          """
          return self.palindromic_lacunas_study(f=f)[1]
  
      def lengths_unioccurrent_lps(self, f=None):
          r"""
          Return the list of the lengths of the unioccurrent longest
          (``f``)-palindromic suffixes (lps) for each non-empty prefix of ``self.`` No
          unioccurrent lps are indicated by ``None``.
  
          It corresponds to the function `H_w` defined in [1] and [2].
  
          INPUT:
  
          - ``f`` -- involution (default: ``None``) on the alphabet of ``self``. It must
            be callable on letters as well as words (e.g. ``WordMorphism``). The
            default value corresponds to usual palindromes, i.e., ``f`` equal to
            the identity.
  
          OUTPUT:
  
          a list -- list of the length of the unioccurrent longest palindromic
          suffix (lps) for each non-empty prefix of ``self``.
          No unioccurrent lps are indicated by ``None``.
  
          EXAMPLES::
  
              sage: w = Word([0,1,1,2,3,4,5,1,13,3])
              sage: w.lengths_unioccurrent_lps()
              [1, 1, 2, 1, 1, 1, 1, None, 1, None]
              sage: f = words.FibonacciWord()[:20]
              sage: f.lengths_unioccurrent_lps() == f.lengths_lps()
              True
              sage: t = words.ThueMorseWord()
              sage: t[:20].lengths_unioccurrent_lps()
              [1, 1, 2, 4, 3, 3, 2, 4, None, None, 6, 8, 10, 12, 14, 16, 6, 8, 10, 12]
              sage: f = WordMorphism({1:[0],0:[1]})
              sage: t[:15].lengths_unioccurrent_lps(f)
              [None, 2, None, 2, None, 4, 6, 8, 4, 6, 4, 6, None, 4, 6]
  
          REFERENCES:
  
          -   [1] A. Blondin-Massé, S. Brlek, S. Labbé, Palindromic lacunas of
              the Thue-Morse word, Proc. GASCOM 2008 (June 16-20 2008, Bibbiena,
              Arezzo-Italia), 53--67.
          -   [2] A. Blondin-Massé, S. Brlek, A. Frosini, S. Labbé, S. Rinaldi,
              Reconstructing words from a fixed palindromic length sequence,
              Proc. TCS 2008, 5th IFIP International Conference on Theoretical
              Computer Science (September 8-10 2008, Milano, Italia), accepted.
          """
          l = self.lengths_lps(f=f)
          for i in self.lacunas(f=f):
              l[i] = None
          return l
  
      def length_maximal_palindrome(self, j, m=None, f=None):
          r"""
          Return the length of the longest palindrome centered at position ``j``.
  
          INPUT:
  
          - ``j`` -- rational, position of the symmetry axis of the palindrome.
            Must return an integer when doubled. It is an integer when the
            center of the palindrome is a letter.
  
          - ``m`` -- integer (default: ``None``), minimal length of palindrome, if known.
            The parity of ``m`` can't be the same as the parity of ``2j``.
  
          - ``f`` -- involution (default: ``None``), on the alphabet. It must be
            callable on letters as well as words (e.g. ``WordMorphism``).
  
          OUTPUT:
  
          length of the longest ``f``-palindrome centered at position ``j``
  
          EXAMPLES::
  
              sage: Word('01001010').length_maximal_palindrome(3/2)
              0
              sage: Word('01101001').length_maximal_palindrome(3/2)
              4
              sage: Word('01010').length_maximal_palindrome(j=3, f='0->1,1->0')
              0
              sage: Word('01010').length_maximal_palindrome(j=2.5, f='0->1,1->0')
              4
              sage: Word('0222220').length_maximal_palindrome(3, f='0->1,1->0,2->2')
              5
  
          ::
  
              sage: w = Word('abcdcbaxyzzyx')
              sage: w.length_maximal_palindrome(3)
              7
              sage: w.length_maximal_palindrome(3, 3)
              7
              sage: w.length_maximal_palindrome(3.5)
              0
              sage: w.length_maximal_palindrome(9.5)
              6
              sage: w.length_maximal_palindrome(9.5, 2)
              6
  
          TESTS:
  
          These are wrong inputs::
  
              sage: w.length_maximal_palindrome(9.6)
              Traceback (most recent call last):
              ...
              ValueError: j must be positive, inferior to length of self
              sage: w.length_maximal_palindrome(3, 2)
              Traceback (most recent call last):
              ...
              ValueError: (2*j-m-1)/2(=3/2) must be an integer, i.e., 2*j(=6) and
              m(=2) can't have the same parity
              sage: w.length_maximal_palindrome(9.5, 3)
              Traceback (most recent call last):
              ...
              ValueError: (2*j-m-1)/2(=15/2) must be an integer, i.e., 2*j(=19) and
              m(=3) can't have the same parity
  
          """
          # Ensure `f` is an involutory word morphism
          if f is not None:
              from sage.combinat.words.morphism import WordMorphism
              if not isinstance(f, WordMorphism):
                  f = WordMorphism(f)
              if not f.is_involution():
                  raise ValueError("f must be an involution")
  
          # Ensure j is a valid entry
          jj = 2*j
          if not jj.is_integer() or j < 0 or j >= len(self):
              raise ValueError("j must be positive, inferior to length of self")
          jj = Integer(jj)
  
          # Initialize length of the known palindrome
          if m is None:
              m = 0 if jj % 2 == 1 else -1
  
          # Initialize the next (left) position to check
          i = (jj - m - 1) / 2
          if not i.is_integer():
              raise ValueError("(2*j-m-1)/2(={}) must be an integer, i.e., "
                               "2*j(={}) and m(={}) can't "
                               "have the same parity".format(i, jj, m))
          i = Integer(i)
  
          # Compute
          if f is None:
              while i >= 0 and jj-i < len(self) and self[i] == self[jj-i]:
                  i -= 1
          else:
              while i >= 0 and jj-i < len(self) and self[i] == f(self[jj-i])[0]:
                  i -= 1
          if jj == 2 * i:
              return 0
          else:
              return jj - 2*i - 1
  
      def lengths_maximal_palindromes(self, f=None):
          r"""
          Return the length of maximal palindromes centered at each position.
  
          INPUT:
  
          - ``f`` -- involution (default: ``None``) on the alphabet of ``self``. It must
            be callable on letters as well as words (e.g. ``WordMorphism``).
  
          OUTPUT:
  
          a list -- The length of the maximal palindrome (or ``f``-palindrome)
          with a given symmetry axis (letter or space between two letters).
  
          EXAMPLES::
  
              sage: Word('01101001').lengths_maximal_palindromes()
              [0, 1, 0, 1, 4, 1, 0, 3, 0, 3, 0, 1, 4, 1, 0, 1, 0]
              sage: Word('00000').lengths_maximal_palindromes()
              [0, 1, 2, 3, 4, 5, 4, 3, 2, 1, 0]
              sage: Word('0').lengths_maximal_palindromes()
              [0, 1, 0]
              sage: Word('').lengths_maximal_palindromes()
              [0]
              sage: Word().lengths_maximal_palindromes()
              [0]
              sage: f = WordMorphism('a->b,b->a')
              sage: Word('abbabaab').lengths_maximal_palindromes(f)
              [0, 0, 2, 0, 0, 0, 2, 0, 8, 0, 2, 0, 0, 0, 2, 0, 0]
          """
          if f is not None :
              from sage.combinat.words.morphism import WordMorphism
              if not isinstance(f, WordMorphism):
                  f = WordMorphism(f)
              if not f.is_involution():
                  raise ValueError("f must be an involution")
  
          LPC = []  # lengths of the maximal palindromes centered at a position
          LPC.append(0)
  
          k = 0  # index, center of rightmost-ending `f`-palindrome encountered
  
          for j in range(1, 2 * len(self) + 1):
              if j >= k + LPC[k]:
                  p = self.length_maximal_palindrome((j - 1)*0.5, -(j%2), f)
                  LPC.append(p)
                  if j + p > k + LPC[k]:
                      k = j
  
              # If the center is included in an encountered `f`-palindrome
              else:
                  # If the `f`-palindrome centered at position j is not the
                  # longest proper `f`-palindromic suffix of the maximal
                  # `f`-palindrome centered at k
                  if LPC[k] + k - j != LPC[2*k - j]:
                      LPC.append(min(LPC[k] + k - j, LPC[2*k - j]))
  
                  else:
                      mp = LPC[k] + k - j
                      p = self.length_maximal_palindrome((j-1)*0.5, mp, f)
                      LPC.append(p)
                      k = j
          return LPC
  
      def lps_lengths(self, f=None):
          r"""
          Return the length of the longest palindromic suffix of each prefix.
  
          INPUT:
  
          - ``f`` -- involution (default: ``None``) on the alphabet of ``self``. It must
            be callable on letters as well as words (e.g. ``WordMorphism``).
  
          OUTPUT:
  
          a list -- The length of the longest palindromic (or ``f``-palindromic)
          suffix of each prefix of ``self``.
  
          EXAMPLES::
  
              sage: Word('01101001').lps_lengths()
              [0, 1, 1, 2, 4, 3, 3, 2, 4]
              sage: Word('00000').lps_lengths()
              [0, 1, 2, 3, 4, 5]
              sage: Word('0').lps_lengths()
              [0, 1]
              sage: Word('').lps_lengths()
              [0]
              sage: Word().lps_lengths()
              [0]
              sage: f = WordMorphism('a->b,b->a')
              sage: Word('abbabaab').lps_lengths(f)
              [0, 0, 2, 0, 2, 2, 4, 6, 8]
          """
          LPC = self.lengths_maximal_palindromes(f)
          LPS = []  # lengths of the longest palindromic suffix of prefixes
          k = 0
          LPS.append(0)
  
          for j in range(1, 2*len(self)+1):
              if j + LPC[j] > k + LPC[k]:
                  for i in range(k + LPC[k] + 1, j + LPC[j] + 1):
                      if i % 2 == 0:
                          LPS.append(i-j)
                      k = j
          return LPS
  
      def palindromes(self, f=None):
          r"""
          Return the set of all palindromic (or ``f``-palindromic) factors of ``self``.
  
          INPUT:
  
          - ``f`` -- involution (default: ``None``) on the alphabet of ``self``. It must
            be callable on letters as well as words (e.g. ``WordMorphism``).
  
          OUTPUT:
  
          a set -- If ``f`` is ``None``, the set of all palindromic factors of ``self``;
          otherwise, the set of all ``f``-palindromic factors of ``self``.
  
          EXAMPLES::
  
              sage: sorted(Word('01101001').palindromes())
              [word: , word: 0, word: 00, word: 010, word: 0110, word: 1, word: 1001, word: 101, word: 11]
              sage: sorted(Word('00000').palindromes())
              [word: , word: 0, word: 00, word: 000, word: 0000, word: 00000]
              sage: sorted(Word('0').palindromes())
              [word: , word: 0]
              sage: sorted(Word('').palindromes())
              [word: ]
              sage: sorted(Word().palindromes())
              [word: ]
              sage: f = WordMorphism('a->b,b->a')
              sage: sorted(Word('abbabaab').palindromes(f))
              [word: , word: ab, word: abbabaab, word: ba, word: baba, word: bbabaa]
          """
          LPS = self.lps_lengths(f)
          return set(self[i-LPS[i] : i] for i in range(len(self)+1))
  
      def palindrome_prefixes(self):
          r"""
          Return a list of all palindrome prefixes of ``self``.
  
          OUTPUT:
  
          a list -- A list of all palindrome prefixes of ``self``.
  
          EXAMPLES::
  
              sage: w = Word('abaaba')
              sage: w.palindrome_prefixes()
              [word: , word: a, word: aba, word: abaaba]
              sage: w = Word('abbbbbbbbbb')
              sage: w.palindrome_prefixes()
              [word: , word: a]
          """
          return list(self.palindrome_prefixes_iterator())
  
      def defect(self, f=None):
          r"""
          Return the defect of ``self``.
  
          The *defect* of a finite word `w` is given by the difference between
          the maximum number of possible palindromic factors in a word of length
          `|w|` and the actual number of palindromic factors contained in `w`.
          It is well known that the maximum number of palindromic factors in `w`
          is `|w|+1` (see [DJP01]_).
  
          An optional involution on letters ``f`` can be given. In that case, the
          *f-palindromic defect* (or *pseudopalindromic defect*, or
          *theta-palindromic defect*) of `w` is returned. It is a
          generalization of defect to f-palindromes. More precisely, the defect is
          `D(w)=|w|+1-g_f(w)-|PAL_f(w)|`, where `PAL_f(w)` denotes the set of
          f-palindromic factors of `w` (including the empty word) and `g_f(w)` is
          the number of pairs `\{a, f(a)\}` such that `a` is a letter, `a` is not
          equal to `f(a)`, and `a` or `f(a)` occurs in `w`. In the case of usual
          palindromes (i.e., for ``f`` not given or equal to the identity),
          `g_f(w) = 0` for all `w`. See [BHNR04]_ for usual palindromes and [Sta11]_
          for f-palindromes.
  
          INPUT:
  
          - ``f`` -- involution (default: ``None``) on the alphabet of ``self``. It must
            be callable on letters as well as words (e.g. ``WordMorphism``). The
            default value corresponds to usual palindromes, i.e., ``f`` equal to
            the identity.
  
          OUTPUT:
  
          an integer -- If ``f`` is ``None``, the palindromic defect of ``self``;
          otherwise, the ``f``-palindromic defect of ``self``.
  
          EXAMPLES::
  
              sage: Word('ara').defect()
              0
              sage: Word('abcacba').defect()
              1
  
          It is known that Sturmian words (see [DJP01]_) have zero defect::
  
              sage: words.FibonacciWord()[:100].defect()
              0
  
              sage: sa = WordMorphism('a->ab,b->b')
              sage: sb = WordMorphism('a->a,b->ba')
              sage: w = (sa*sb*sb*sa*sa*sa*sb).fixed_point('a')
              sage: w[:30].defect()
              0
              sage: w[110:140].defect()
              0
  
          It is even conjectured that the defect of an aperiodic word which is
          a fixed point of a primitive morphism is either `0` or infinite
          (see [BBGL08]_)::
  
              sage: w = words.ThueMorseWord()
              sage: w[:50].defect()
              12
              sage: w[:100].defect()
              16
              sage: w[:300].defect()
              52
  
          For generalized defect with an involution different from the identity,
          there is always a letter which is not a palindrome! This is the reason
          for the modification of the definition::
  
              sage: f = WordMorphism('a->b,b->a')
              sage: Word('a').defect(f)
              0
              sage: Word('ab').defect(f)
              0
              sage: Word('aa').defect(f)
              1
              sage: Word('abbabaabbaababba').defect(f)
              3
  
          ::
  
              sage: f = WordMorphism('a->b,b->a,c->c')
              sage: Word('cabc').defect(f)
              0
              sage: Word('abcaab').defect(f)
              2
  
          Other examples::
  
              sage: Word('000000000000').defect()
              0
              sage: Word('011010011001').defect()
              2
              sage: Word('0101001010001').defect()
              0
              sage: Word().defect()
              0
              sage: Word('abbabaabbaababba').defect()
              2
  
          REFERENCES:
  
          .. [BBGL08] \A. Blondin Massé, S. Brlek, A. Garon, and S. Labbé,
             Combinatorial properties of f -palindromes in the Thue-Morse
             sequence. Pure Math. Appl., 19(2-3):39--52, 2008.
  
          .. [BHNR04] \S. Brlek, S. Hamel, M. Nivat, C. Reutenauer, On the
             Palindromic Complexity of Infinite Words, in J. Berstel, J.
             Karhumaki, D. Perrin, Eds, Combinatorics on Words with Applications,
             International Journal of Foundation of Computer Science, Vol. 15,
             No. 2 (2004) 293--306.
  
          .. [DJP01] \X. Droubay, J. Justin, G. Pirillo, Episturmian words and some
             constructions of de Luca and Rauzy, Theoret. Comput. Sci. 255,
             (2001), no. 1--2, 539--553.
  
          .. [Sta11] \Š. Starosta, On Theta-palindromic Richness, Theoret. Comp.
             Sci. 412 (2011) 1111--1121
          """
          g_w = 0
          if f is not None:
              from sage.combinat.words.morphism import WordMorphism
              if not isinstance(f, WordMorphism):
                  f = WordMorphism(f)
              if not f.is_involution():
                  raise ValueError("f must be an involution")
              D = f.domain()
              A = set(map(D, self.letters()))
              while A:
                  x = A.pop()
                  if f(x) != x: # count only non f-palindromic letters
                      if f(x) in A:
                          A.remove(f(x))
                      g_w +=1
  
          return self.length()+1-g_w-len(self.palindromes(f=f))
  
      def is_full(self, f=None):
          r"""
          Return ``True`` if ``self`` has defect `0`, and ``False`` otherwise.
  
          A word is *full* (or *rich*) if its defect is zero (see [1]).
          If ``f`` is given, then the ``f``-palindromic defect is used (see [2]).
  
          INPUT:
  
          - ``f`` -- involution (default: ``None``) on the alphabet of ``self``. It must
            be callable on letters as well as words (e.g. ``WordMorphism``).
  
          OUTPUT:
  
          boolean -- If ``f`` is ``None``, whether ``self`` is full;
          otherwise, whether ``self`` is full of ``f``-palindromes.
  
          EXAMPLES::
  
              sage: words.ThueMorseWord()[:100].is_full()
              False
              sage: words.FibonacciWord()[:100].is_full()
              True
              sage: Word('000000000000000').is_full()
              True
              sage: Word('011010011001').is_full()
              False
              sage: Word('2194').is_full()
              True
              sage: Word().is_full()
              True
  
          ::
  
              sage: f = WordMorphism('a->b,b->a')
              sage: Word().is_full(f)
              True
              sage: w = Word('ab')
              sage: w.is_full()
              True
              sage: w.is_full(f)
              True
  
          ::
  
              sage: f = WordMorphism('a->b,b->a')
              sage: Word('abab').is_full(f)
              True
              sage: Word('abba').is_full(f)
              False
  
          A simple example of an infinite word full of f-palindromes::
  
              sage: p = WordMorphism({0:'abc',1:'ab'})
              sage: f = WordMorphism('a->b,b->a,c->c')
              sage: p(words.FibonacciWord()[:50]).is_full(f)
              True
              sage: p(words.FibonacciWord()[:150]).is_full(f)
              True
  
          REFERENCES:
  
          -   [1] S. Brlek, S. Hamel, M. Nivat, C. Reutenauer, On the Palindromic
              Complexity of Infinite Words, in J. Berstel, J. Karhumaki,
              D. Perrin, Eds, Combinatorics on Words with Applications,
              International Journal of Foundation of Computer Science, Vol. 15,
              No. 2 (2004) 293--306.
  
          -   [2] E. Pelantová, Š. Starosta, Infinite words rich and almost rich
              in generalized palindromes, in: G. Mauri, A. Leporati (Eds.),
              Developments in Language Theory, volume 6795 of Lecture Notes
              in Computer Science, Springer-Verlag, Berlin, Heidelberg, 2011,
              pp. 406--416
          """
          return self.defect(f=f) == 0
  
      is_rich = is_full
  
      def palindromic_closure(self, side='right', f=None):
          r"""
          Return the shortest palindrome having ``self`` as a prefix
          (or as a suffix if ``side`` is ``'left'``).
  
          See [1].
  
          INPUT:
  
          - ``side`` -- ``'right'`` or ``'left'`` (default: ``'right'``) the
            direction of the  closure
  
          - ``f`` -- involution (default: ``None``) on the alphabet of ``self``.
            It must be callable on letters as well as words (e.g. ``WordMorphism``).
  
          OUTPUT:
  
          a word -- If ``f`` is ``None``, the right palindromic closure of ``self``;
          otherwise, the right ``f``-palindromic closure of ``self``.
          If ``side`` is ``'left'``, the left palindromic closure.
  
          EXAMPLES::
  
              sage: Word('1233').palindromic_closure()
              word: 123321
              sage: Word('12332').palindromic_closure()
              word: 123321
              sage: Word('0110343').palindromic_closure()
              word: 01103430110
              sage: Word('0110343').palindromic_closure(side='left')
              word: 3430110343
              sage: Word('01105678').palindromic_closure(side='left')
              word: 876501105678
              sage: w = Word('abbaba')
              sage: w.palindromic_closure()
              word: abbababba
  
          ::
  
              sage: f = WordMorphism('a->b,b->a')
              sage: w.palindromic_closure(f=f)
              word: abbabaab
              sage: w.palindromic_closure(f=f, side='left')
              word: babaabbaba
  
          TESTS::
  
              sage: f = WordMorphism('a->c,c->a')
              sage: w.palindromic_closure(f=f, side='left')
              Traceback (most recent call last):
              ...
              KeyError: 'b'
  
          REFERENCES:
  
          -   [1] A. de Luca, A. De Luca, Pseudopalindrome closure operators
              in free monoids, Theoret. Comput. Sci. 362 (2006) 282--300.
          """
          if f is None:
              if side == 'right':
                  l = self.lps().length()
                  #return self * self[-(l+1)::-1]
                  return self * self[:self.length()-l].reversal()
              elif side == 'left':
                  l = self.reversal().lps().length()
                  return self[:l-1:-1] * self
              else:
                  raise ValueError("side must be either 'left' or 'right' (not %s) " % side)
          else:
              from sage.combinat.words.morphism import WordMorphism
              f = WordMorphism(f)
              if not f.is_involution():
                  raise ValueError("f must be an involution")
              if side == 'right':
                  l = self.lps(f=f).length()
                  return self * f(self[-(l+1)::-1])
              elif side == 'left':
                  l = self.reversal().lps(f=f).length()
                  return f(self[:l-1:-1]) * self
              else:
                  raise ValueError("side must be either 'left' or 'right' (not %s) " % side)
  
      def is_symmetric(self, f=None):
          r"""
          Return ``True`` if ``self`` is symmetric (or ``f``-symmetric), and
          ``False`` otherwise.
  
          A word is *symmetric* (resp. `f`-*symmetric*) if it is the
          product of two palindromes (resp. `f`-palindromes). See [1] and [2].
  
          INPUT:
  
          - ``f`` -- involution (default: ``None``) on the alphabet of ``self``. It must
            be callable on letters as well as words (e.g. ``WordMorphism``).
  
          EXAMPLES::
  
              sage: Word('abbabab').is_symmetric()
              True
              sage: Word('ababa').is_symmetric()
              True
              sage: Word('aababaabba').is_symmetric()
              False
              sage: Word('aabbbaababba').is_symmetric()
              False
              sage: f = WordMorphism('a->b,b->a')
              sage: Word('aabbbaababba').is_symmetric(f)
              True
  
          REFERENCES:
  
          -   [1] S. Brlek, S. Hamel, M. Nivat, C. Reutenauer, On the Palindromic
              Complexity of Infinite Words, in J. Berstel, J. Karhumaki,
              D. Perrin, Eds, Combinatorics on Words with Applications,
              International Journal of Foundation of Computer Science, Vol. 15,
              No. 2 (2004) 293--306.
          -   [2] A. de Luca, A. De Luca, Pseudopalindrome closure operators
              in free monoids, Theoret. Comput. Sci. 362 (2006) 282--300.
          """
  
          square = self*self
          return square.lps_lengths(f)[-1] >= self.length()
  
      def length_border(self):
          r"""
          Return the length of the border of ``self``.
  
          The *border* of a word is the longest word that is both a proper
          prefix and a proper suffix of ``self``.
  
          EXAMPLES::
  
              sage: Word('121').length_border()
              1
              sage: Word('1').length_border()
              0
              sage: Word('1212').length_border()
              2
              sage: Word('111').length_border()
              2
              sage: Word().length_border() is None
              True
          """
          if self.is_empty():
              return None
          return self.prefix_function_table()[-1]
  
      def border(self):
          r"""
          Return the longest word that is both a proper prefix and a proper
          suffix of ``self``.
  
          EXAMPLES::
  
              sage: Word('121212').border()
              word: 1212
              sage: Word('12321').border()
              word: 1
              sage: Word().border() is None
              True
          """
          if self.is_empty():
              return None
          return self[:self.length_border()]
  
      def minimal_period(self):
          r"""
          Return the period of ``self``.
  
          Let `A` be an alphabet. An integer `p\geq 1` is a *period* of a
          word `w=a_1a_2\cdots a_n` where `a_i\in A` if `a_i=a_{i+p}` for
          `i=1,\ldots,n-p`. The smallest period of `w` is called *the*
          period of `w`. See Chapter 1 of [1].
  
          EXAMPLES::
  
              sage: Word('aba').minimal_period()
              2
              sage: Word('abab').minimal_period()
              2
              sage: Word('ababa').minimal_period()
              2
              sage: Word('ababaa').minimal_period()
              5
              sage: Word('ababac').minimal_period()
              6
              sage: Word('aaaaaa').minimal_period()
              1
              sage: Word('a').minimal_period()
              1
              sage: Word().minimal_period()
              1
  
          REFERENCES:
  
          -   [1] M. Lothaire, Algebraic Combinatorics On Words, vol. 90 of
              Encyclopedia of Mathematics and its Applications, Cambridge
              University Press, U.K., 2002.
          """
          if self.is_empty():
              return 1
          return self.length()-self.length_border()
  
      def order(self):
          r"""
          Return the order of ``self``.
  
          Let `p(w)` be the period of a word `w`. The positive rational number
          `|w|/p(w)` is the *order* of `w`. See Chapter 8 of [1].
  
          OUTPUT:
  
          rational -- the order
  
          EXAMPLES::
  
              sage: Word('abaaba').order()
              2
              sage: Word('ababaaba').order()
              8/5
              sage: Word('a').order()
              1
              sage: Word('aa').order()
              2
              sage: Word().order()
              0
  
          REFERENCES:
  
          -   [1] M. Lothaire, Algebraic Combinatorics On Words, vol. 90 of
              Encyclopedia of Mathematics and its Applications, Cambridge
              University Press, U.K., 2002.
          """
          from sage.rings.rational import Rational
          return Rational((self.length(),self.minimal_period()))
  
      def critical_exponent(self):
          r"""
          Return the critical exponent of ``self``.
  
          The *critical exponent* of a word is the supremum of the order of
          all its (finite) factors. See [1].
  
          .. note::
  
              The implementation here uses the suffix tree to enumerate all the
              factors. It should be improved.
  
          EXAMPLES::
  
              sage: Word('aaba').critical_exponent()
              2
              sage: Word('aabaa').critical_exponent()
              2
              sage: Word('aabaaba').critical_exponent()
              7/3
              sage: Word('ab').critical_exponent()
              1
              sage: Word('aba').critical_exponent()
              3/2
              sage: words.ThueMorseWord()[:20].critical_exponent()
              2
  
          REFERENCES:
  
          .. [Dejean] \F. Dejean. Sur un théorème de Thue. J. Combinatorial Theory
             Ser. A 13:90--99, 1972.
          """
          return max(map(FiniteWord_class.order, self.factor_iterator()))
  
      def is_overlap(self):
          r"""
          Return ``True`` if ``self`` is an overlap, and ``False`` otherwise.
  
          EXAMPLES::
  
              sage: Word('12121').is_overlap()
              True
              sage: Word('123').is_overlap()
              False
              sage: Word('1231').is_overlap()
              False
              sage: Word('123123').is_overlap()
              False
              sage: Word('1231231').is_overlap()
              True
              sage: Word().is_overlap()
              False
          """
          if self.length() == 0:
              return False
          return self.length_border() > self.length()//2
  
      def primitive_length(self):
          r"""
          Return the length of the primitive of ``self``.
  
          EXAMPLES::
  
              sage: Word('1231').primitive_length()
              4
              sage: Word('121212').primitive_length()
              2
          """
          l = lu = self.length()
          if l == 0:
              return 0
          p = self.prefix_function_table()
          while l > 0:
              l = p[l-1]
              if lu % (lu - l) == 0:
                  return lu - l
  
      def is_primitive(self):
          r"""
          Return ``True`` if ``self`` is primitive, and ``False`` otherwise.
  
          A finite word `w` is *primitive* if it is not a positive integer
          power of a shorter word.
  
          EXAMPLES::
  
              sage: Word('1231').is_primitive()
              True
              sage: Word('111').is_primitive()
              False
          """
          return self.length() == self.primitive_length()
  
      def primitive(self):
          r"""
          Return the primitive of ``self``.
  
          EXAMPLES::
  
              sage: Word('12312').primitive()
              word: 12312
              sage: Word('121212').primitive()
              word: 12
          """
          return self[:self.primitive_length()]
  
      def exponent(self):
          r"""
          Return the exponent of ``self``.
  
          OUTPUT:
  
          integer -- the exponent
  
          EXAMPLES::
  
              sage: Word('1231').exponent()
              1
              sage: Word('121212').exponent()
              3
              sage: Word().exponent()
              0
          """
          if self.length() == 0:
              return 0
          return self.length() // self.primitive_length()
  
      def has_period(self, p):
          r"""
          Return ``True`` if ``self`` has the period ``p``,
          ``False`` otherwise.
  
          .. NOTE::
  
              By convention, integers greater than the length
              of ``self`` are periods of ``self``.
  
          INPUT:
  
          - ``p`` -- an integer to check if it is a period
            of ``self``.
  
          EXAMPLES::
  
              sage: w = Word('ababa')
              sage: w.has_period(2)
              True
              sage: w.has_period(3)
              False
              sage: w.has_period(4)
              True
              sage: w.has_period(-1)
              False
              sage: w.has_period(5)
              True
              sage: w.has_period(6)
              True
          """
          if p < 0:
              return False
          elif p >= len(self):
              return True
          else:
              for i in range(len(self) - p):
                  if self[i] != self[i + p]:
                      return False
              return True
  
      def periods(self, divide_length=False):
          r"""
          Return a list containing the periods of ``self``
          between `1` and `n - 1`, where `n` is the length
          of ``self``.
  
          INPUT:
  
          - ``divide_length`` -- boolean (default: ``False``).
            When set to ``True``, then only periods that divide
            the length of ``self`` are considered.
  
          OUTPUT:
  
          a list of positive integers
  
          EXAMPLES::
  
              sage: w = Word('ababab')
              sage: w.periods()
              [2, 4]
              sage: w.periods(divide_length=True)
              [2]
              sage: w = Word('ababa')
              sage: w.periods()
              [2, 4]
              sage: w.periods(divide_length=True)
              []
          """
          n = len(self)
          if divide_length:
              possible = (i for i in range(1,n) if n % i == 0)
          else:
              possible = range(1, n)
          return [x for x in possible if self.has_period(x)]
  
      def longest_common_subword(self, other):
          r"""
          Return a longest subword of ``self`` and ``other``.
  
          A subword of a word is a subset of the word's letters, read in the
          order in which they appear in the word.
  
          For more information, see
          :wikipedia:`Longest_common_subsequence_problem`.
  
          INPUT:
  
          - ``other`` -- a word
  
          ALGORITHM:
  
          For any indices `i,j`, we compute the longest common subword ``lcs[i,j]`` of
          ``self[:i]`` and ``other[:j]``. This can be easily obtained as the longest
          of
  
          - ``lcs[i-1,j]``
  
          - ``lcs[i,j-1]``
  
          - ``lcs[i-1,j-1]+self[i]`` if ``self[i]==other[j]``
  
          EXAMPLES::
  
              sage: v1 = Word("abc")
              sage: v2 = Word("ace")
              sage: v1.longest_common_subword(v2)
              word: ac
  
              sage: w1 = Word("1010101010101010101010101010101010101010")
              sage: w2 = Word("0011001100110011001100110011001100110011")
              sage: w1.longest_common_subword(w2)
              word: 00110011001100110011010101010
  
          TESTS::
  
              sage: Word().longest_common_subword(Word())
              word:
  
          .. SEEALSO::
  
              :meth:`is_subword_of`
          """
          from sage.combinat.words.word import Word
          if len(self) == 0 or len(other) == 0:
              return Word()
  
          w2 = list(other)
  
          # In order to avoid storing lcs[i,j] for each pair i,j of indices, we
          # only store the lcs[i,j] for two consecutive values of i. At any step
          # of the algorithm, lcs[i,j] is stored at lcs[0][j] and lcs[i-1,j] is
          # stored at lcs[1][j]
  
          # The weird +1 that follows exists to make sure that lcs[i,-1] returns
          # the empty word.
          lcs = [[[] for i in range(len(w2)+1)] for j in range(2)]
  
          for i,l1 in enumerate(self):
              for j,l2 in enumerate(other):
                  lcs[0][j] = max(lcs[0][j-1], lcs[1][j],
                                  lcs[1][j-1] + ([l1] if l1==l2 else []),key=len)
  
              # Maintaining the meaning of lcs for the next loop
              lcs.pop(1)
              lcs.insert(0,[[] for i in range(len(w2)+1)])
  
          return Word(lcs[1][-2])
  
      def is_subword_of(self, other):
          r"""
          Return ``True`` is ``self`` is a subword of ``other``, and ``False`` otherwise.
  
          EXAMPLES::
  
              sage: Word().is_subword_of(Word('123'))
              True
              sage: Word('123').is_subword_of(Word('3211333213233321'))
              True
              sage: Word('321').is_subword_of(Word('11122212112122133111222332'))
              False
  
          .. SEEALSO::
  
              :meth:`longest_common_subword`
  
          """
          its = iter(self)
          try:
              s = next(its)
              for e in other:
                  if s == e:
                      s = next(its)
              else:
                  return False
          except StopIteration:
              return True
  
      def is_lyndon(self):
          r"""
          Return ``True`` if ``self`` is a Lyndon word, and ``False``
          otherwise.
  
          A *Lyndon word* is a non-empty word that is lexicographically
          smaller than each of its proper suffixes (for the given order
          on its alphabet). That is, `w` is a Lyndon word if `w` is non-empty
          and for each factorization `w = uv` (with `u`, `v` both non-empty),
          we have `w < v`.
  
          Equivalently, `w` is a Lyndon word iff `w` is a non-empty word that is
          lexicographically smaller than each of its proper conjugates for the
          given order on its alphabet.
  
          See for instance [1].
  
          EXAMPLES::
  
              sage: Word('123132133').is_lyndon()
              True
              sage: Word().is_lyndon()
              False
              sage: Word('122112').is_lyndon()
              False
  
          TESTS:
  
          A sanity check: ``LyndonWords`` generates Lyndon words, so we
          filter all words of length `n<10` on the alphabet [1,2,3] for
          Lyndon words, and compare with the ``LyndonWords`` generator::
  
              sage: for n in range(1,10):
              ....:     lw1 = [w for w in Words([1,2,3], n) if w.is_lyndon()]
              ....:     lw2 = LyndonWords(3,n)
              ....:     if set(lw1) != set(lw2): print(False)
  
          Filter all words of length 8 on the alphabet [c,a,b] for Lyndon
          words, and compare with the :class:`LyndonWords` generator after
          mapping [a,b,c] to [2,3,1]::
  
              sage: lw = [w for w in Words('cab', 8) if w.is_lyndon()]
              sage: phi = WordMorphism({'a':2,'b':3,'c':1})
              sage: set(map(phi, lw)) == set(LyndonWords(3,8))
              True
  
          REFERENCES:
  
          -   [1] M. Lothaire, Combinatorics On Words, vol. 17 of Encyclopedia
              of Mathematics and its Applications, Addison-Wesley, Reading,
              Massachusetts, 1983.
  
          """
          if self.is_empty():
              return False
          key = self.parent().sortkey_letters
          n = self.length()
          i, j = 0, 1
          while j < n:
              ki = key(self[i])
              kj = key(self[j])
              if ki == kj:
                  # increment i and j
                  i += 1
                  j += 1
              elif ki < kj:
                  # reset i, increment j
                  i = 0
                  j += 1
              else:
                  # we found the first word in the lyndon factorization;
                  return False
          else:
              return i == 0
  
      def lyndon_factorization(self):
          r"""
          Return the Lyndon factorization of ``self``.
  
          The *Lyndon factorization* of a finite word `w` is the unique
          factorization of `w` as a non-increasing product of Lyndon words,
          i.e., `w = l_1\cdots l_n` where each `l_i` is a Lyndon word and
          `l_1\geq \cdots \geq l_n`. See for instance [1].
  
          OUTPUT:
  
          the list `[l_1, \ldots, l_n]` of factors obtained
  
          EXAMPLES::
  
              sage: Word('010010010001000').lyndon_factorization()
              (01, 001, 001, 0001, 0, 0, 0)
              sage: Words('10')('010010010001000').lyndon_factorization()
              (0, 10010010001000)
              sage: Word('abbababbaababba').lyndon_factorization()
              (abb, ababb, aababb, a)
              sage: Words('ba')('abbababbaababba').lyndon_factorization()
              (a, bbababbaaba, bba)
              sage: Word([1,2,1,3,1,2,1]).lyndon_factorization()
              (1213, 12, 1)
  
          TESTS::
  
              sage: Words('01')('').lyndon_factorization()
              ()
              sage: Word('01').lyndon_factorization()
              (01)
              sage: Words('10')('01').lyndon_factorization()
              (0, 1)
              sage: lynfac = Word('abbababbaababba').lyndon_factorization()
              sage: [x.is_lyndon() for x in lynfac]
              [True, True, True, True]
              sage: lynfac = Words('ba')('abbababbaababba').lyndon_factorization()
              sage: [x.is_lyndon() for x in lynfac]
              [True, True, True]
              sage: w = words.ThueMorseWord()[:1000]
              sage: w == prod(w.lyndon_factorization())
              True
  
          REFERENCES:
  
          -   [1] J.-P. Duval, Factorizing words over an ordered alphabet,
              J. Algorithms 4 (1983) 363--381.
  
          -   [2] G. Melancon, Factorizing infinite words using Maple,
              MapleTech journal, vol. 4, no. 1, 1997, pp. 34-42.
  
          """
          key = self.parent().sortkey_letters
          # We compute the indexes of the factorization.
          n = self.length()
          k = -1
          F = [0]
          while k < n-1:
              i = k+1
              j = k+2
              while j < n:
                  ki = key(self[i])
                  kj = key(self[j])
                  if ki < kj:
                      i = k+1
                      j += 1
                  elif ki == kj:
                      i += 1
                      j += 1
                  else:
                      break
              while k < i:
                  F.append(k + j - i + 1)
                  k = k + j - i
          return Factorization([self[F[i]:F[i+1]] for i in range(len(F)-1)])
  
      def inversions(self):
          r"""
          Return a list of the inversions of ``self``. An inversion is a pair
          `(i,j)` of non-negative integers `i < j` such that ``self[i] > self[j]``.
  
          EXAMPLES::
  
              sage: Word([1,2,3,2,2,1]).inversions()
              [[1, 5], [2, 3], [2, 4], [2, 5], [3, 5], [4, 5]]
              sage: Words([3,2,1])([1,2,3,2,2,1]).inversions()
              [[0, 1], [0, 2], [0, 3], [0, 4], [1, 2]]
              sage: Word('abbaba').inversions()
              [[1, 3], [1, 5], [2, 3], [2, 5], [4, 5]]
              sage: Words('ba')('abbaba').inversions()
              [[0, 1], [0, 2], [0, 4], [3, 4]]
          """
          inversion_list = []
          cmp_key = self._parent.sortkey_letters
          for (i1, letter1) in enumerate(self):
              k1 = cmp_key(letter1)
              for (i2, letter2) in enumerate(self[i1 + 1:]):
                  k2 = cmp_key(letter2)
                  if k1 > k2:
                      inversion_list.append([i1, i1 + i2 + 1])
          return inversion_list
  
      # TODO: This function should be defined for words of integers, but it
      # naturally is defined over an alphabet with a rank function....
      def degree(self, weights=None):
          r"""
          Return the weighted degree of ``self``, where the weighted degree of
          each letter in the ordered alphabet is given by ``weights``, which
          defaults to ``[1, 2, 3, ...]``.
  
          INPUT:
  
          -  ``weights`` -- a list or a tuple, or a dictionary keyed by the
             letters occurring in ``self``.
  
          EXAMPLES::
  
              sage: Word([1,2,3]).degree()
              6
              sage: Word([3,2,1]).degree()
              6
              sage: Words("ab")("abba").degree()
              6
              sage: Words("ab")("abba").degree([0,2])
              4
              sage: Words("ab")("abba").degree([-1,-1])
              -4
              sage: Words("ab")("aabba").degree([1,1])
              5
              sage: Words([1,2,4])([1,2,4]).degree()
              6
              sage: Word([1,2,4]).degree()
              7
              sage: Word("aabba").degree({'a':1,'b':2})
              7
              sage: Word([0,1,0]).degree({0:17,1:0})
              34
          """
          if isinstance(weights, dict):
              deg = 0
              for a in self:
                  deg += weights[a]
              return deg
  
          if hasattr(self._parent._alphabet, "rank"):
              rank_fcn = self._parent._alphabet.rank
              deg = 0
              if weights is None:
                  rank = {}
                  for a in self:
                      if a not in rank:
                          rank[a] = rank_fcn(a)
                      deg += rank[a]+1
              elif isinstance(weights, (list,tuple)):
                  rank = {}
                  for a in self:
                      if a not in rank:
                          rank[a] = rank_fcn(a)
                      deg += weights[rank[a]]
              return deg
  
          if all(x in ZZ for x in self):
              return sum(self)
  
          raise TypeError("degree is not defined for your word")
  
      def deg_lex_less(self, other, weights=None):
          r"""
          Return ``True`` if ``self`` is degree lexicographically less than ``other``,
          and ``False`` otherwise. The weight of each letter in the ordered
          alphabet is given by ``weights``, which defaults to ``[1, 2, 3, ...]``.
  
          EXAMPLES::
  
              sage: Word([1,2,3]).deg_lex_less(Word([1,3,2]))
              True
              sage: Word([3,2,1]).deg_lex_less(Word([1,2,3]))
              False
              sage: W = Words(range(5))
              sage: W([1,2,4]).deg_lex_less(W([1,3,2]))
              False
              sage: Word("abba").deg_lex_less(Word("abbb"), dict(a=1,b=2))
              True
              sage: Word("abba").deg_lex_less(Word("baba"), dict(a=1,b=2))
              True
              sage: Word("abba").deg_lex_less(Word("aaba"), dict(a=1,b=2))
              False
              sage: Word("abba").deg_lex_less(Word("aaba"), dict(a=1,b=0))
              True
          """
          deg_self = self.degree(weights)
          deg_other = other.degree(weights)
          if deg_self != deg_other:
              return deg_self < deg_other
          return self.lex_less(other)
  
      def inv_lex_less(self, other):
          r"""
          Return ``True`` if ``self`` is inverse lexicographically less than ``other``.
  
          EXAMPLES::
  
              sage: Word([1,2,4]).inv_lex_less(Word([1,3,2]))
              False
              sage: Word([3,2,1]).inv_lex_less(Word([1,2,3]))
              True
          """
          if self.length() != len(other):
              return self.length() < len(other)
          return self.reversal() < other.reversal()
  
      def deg_inv_lex_less(self, other, weights=None):
          r"""
          Return ``True`` if the word ``self`` is degree inverse lexicographically
          less than ``other``.
  
          EXAMPLES::
  
              sage: Word([1,2,4]).deg_inv_lex_less(Word([1,3,2]))
              False
              sage: Word([3,2,1]).deg_inv_lex_less(Word([1,2,3]))
              True
          """
          d1 = self.degree(weights)
          d2 = other.degree(weights)
          if d1 != d2:
              return d1 < d2
          return self.inv_lex_less(other)
  
      def rev_lex_less(self, other):
          r"""
          Return ``True`` if the word ``self`` is reverse
          lexicographically less than ``other``.
  
          EXAMPLES::
  
              sage: Word([1,2,4]).rev_lex_less(Word([1,3,2]))
              True
              sage: Word([3,2,1]).rev_lex_less(Word([1,2,3]))
              False
          """
          if self.length() != len(other):
              return self.length() > len(other)
          return self.reversal() > other.reversal()
  
      def deg_rev_lex_less(self, other, weights=None):
          r"""
          Return ``True`` if ``self`` is degree reverse
          lexicographically less than ``other``.
  
          EXAMPLES::
  
              sage: Word([3,2,1]).deg_rev_lex_less(Word([1,2,3]))
              False
              sage: Word([1,2,4]).deg_rev_lex_less(Word([1,3,2]))
              False
              sage: Word([1,2,3]).deg_rev_lex_less(Word([1,2,4]))
              True
          """
          d1 = self.degree(weights)
          d2 = other.degree(weights)
          if d1 != d2:
              return d1 < d2
          return self.rev_lex_less(other)
  
      @cached_method
      def last_position_dict(self):
          r"""
          Return a dictionary that contains the last position of each letter
          in ``self``.
  
          EXAMPLES::
  
              sage: Word('1231232').last_position_dict()
              {'1': 3, '2': 6, '3': 5}
          """
          d = {}
          for (i, letter) in enumerate(self):
              d[letter] = i
          return d
  
      def _pos_in(self, other, p):
          r"""
          Return the position of the first occurrence of ``self`` starting at
          position ``p`` in ``other``.
  
          EXAMPLES::
  
              sage: Word('12')._pos_in(Word('131231'), 2)
              2
              sage: Word('12')._pos_in(Word('131231'), 3) is None
              True
              sage: Word('32')._pos_in(Word('131231'), 0) is None
              True
  
          The empty word occurs in a word::
  
              sage: Word('')._pos_in(Word('123'), 0)
              0
              sage: Word('')._pos_in(Word(''), 0)
              0
          """
          lf = self.length()
          lm = other.length()
          if lf == 0:
              return p
          elif lm == 0:
              return None
          occ = self.last_position_dict()
          suff = self.good_suffix_table()
          s = p
          while s <= lm - lf:
              for j in range(lf-1, -1, -1):
                  a = other[s+j]
                  if self[j] != a :
                      s += max(suff[j + 1], j - occ.get(a,-1))
                      break
              else:
                  return s
          return None
  
      def first_pos_in(self, other):
          r"""
          Return the position of the first occurrence of ``self`` in ``other``,
          or ``None`` if ``self`` is not a factor of ``other``.
  
          EXAMPLES::
  
              sage: Word('12').first_pos_in(Word('131231'))
              2
              sage: Word('32').first_pos_in(Word('131231')) is None
              True
          """
          return self._pos_in(other, 0)
  
      def find(self, sub, start=0, end=None):
          r"""
          Return the index of the first occurrence of ``sub`` in ``self``,
          such that ``sub`` is contained within ``self[start:end]``.
          Return ``-1`` on failure.
  
          INPUT:
  
          - ``sub`` -- string, list, tuple or word to search for.
  
          - ``start`` -- non-negative integer (default: ``0``) specifying
            the position from which to start the search.
  
          - ``end`` -- non-negative integer (default: ``None``) specifying
            the position at which the search must stop. If ``None``, then
            the search is performed up to the end of the string.
  
          OUTPUT:
  
          a non-negative integer or ``-1``
  
          EXAMPLES::
  
              sage: w = Word([0,1,0,0,1])
              sage: w.find(Word([1,0]))
              1
  
          The ``sub`` argument can also be a tuple or a list::
  
              sage: w.find([1,0])
              1
              sage: w.find((1,0))
              1
  
          Examples using ``start`` and ``end``::
  
              sage: w.find(Word([0,1]), start=1)
              3
              sage: w.find(Word([0,1]), start=1, end=5)
              3
              sage: w.find(Word([0,1]), start=1, end=4) == -1
              True
              sage: w.find(Word([1,1])) == -1
              True
              sage: w.find("aa")
              -1
  
          Instances of ``Word_str`` handle string inputs as well::
  
              sage: w = Word('abac')
              sage: w.find('a')
              0
              sage: w.find('ba')
              1
  
          TESTS:
  
          Check that :trac:`12804` is fixed::
  
              sage: w = Word(iter("ababab"), length="finite")
              sage: w.find("ab")
              0
              sage: w.find("ab", start=1)
              2
              sage: w.find("aa")
              -1
              sage: w.find("abc")
              -1
              sage: w = Words('ab')(tuple('babaabaaab'))
              sage: w.find('abc')
              -1
  
          """
          if not isinstance(sub, FiniteWord_class):
              try:
                  sub = self.parent()(sub)
              except (ValueError,TypeError):
                  return -1
          p = sub.first_pos_in(self[start:end])
          return -1 if p is None else p+start
  
      def rfind(self, sub, start=0, end=None):
          r"""
          Return the index of the last occurrence of ``sub`` in ``self``,
          such that ``sub`` is contained within ``self[start:end]``.
          Return ``-1`` on failure.
  
          INPUT:
  
          - ``sub`` -- string, list, tuple or word to search for.
  
          - ``start`` -- non-negative integer (default: ``0``) specifying
            the position at which the search must stop.
  
          - ``end`` -- non-negative integer (default: ``None``) specifying
            the position from which to start the search. If ``None``, then
            the search is performed up to the end of the string.
  
          OUTPUT:
  
          a non-negative integer or ``-1``
  
          EXAMPLES::
  
              sage: w = Word([0,1,0,0,1])
              sage: w.rfind(Word([0,1]))
              3
  
          The ``sub`` parameter can also be a list or a tuple::
  
              sage: w.rfind([0,1])
              3
              sage: w.rfind((0,1))
              3
  
          Examples using the argument ``start`` and ``end``::
  
              sage: w.rfind(Word([0,1]), end=4)
              0
              sage: w.rfind(Word([0,1]), end=5)
              3
              sage: w.rfind(Word([0,0]), start=2, end=5)
              2
              sage: w.rfind(Word([0,0]), start=3, end=5)
              -1
  
          Instances of ``Word_str`` handle string inputs as well::
  
              sage: w = Word('abac')
              sage: w.rfind('a')
              2
              sage: w.rfind(Word('a'))
              2
              sage: w.rfind([0,1])
              -1
  
          TESTS:
  
          Check that :trac:`12804` is fixed::
  
              sage: w = Word(iter("abab"), length="finite")
              sage: w.rfind("ab")
              2
              sage: w.rfind("ab", end=3)
              0
              sage: w.rfind("aa")
              -1
              sage: w.rfind([0,0,0])
              -1
          """
          if not isinstance(sub, FiniteWord_class):
              try:
                  sub = self.parent()(sub)
              except (ValueError,TypeError):
                  return -1
          L = len(sub)
          start = max(0, int(start))
          if end is None:
              i = len(self) - L
          else:
              i = min(end, len(self)) - L
          while i >= start:
              if self[i:i+L] == sub: return i
              i -= 1
          return -1
  
      def is_factor(self, other):
          r"""
          Return ``True`` if ``self`` is a factor of ``other``, and ``False`` otherwise.
  
          EXAMPLES::
  
              sage: u = Word('2113')
              sage: w = Word('123121332131233121132123')
              sage: u.is_factor(w)
              True
              sage: u = Word('321')
              sage: w = Word('1231241231312312312')
              sage: u.is_factor(w)
              False
  
          The empty word is factor of another word::
  
              sage: Word().is_factor(Word())
              True
              sage: Word().is_factor(Word('a'))
              True
              sage: Word().is_factor(Word([1,2,3]))
              True
              sage: Word().is_factor(Word(lambda n:n, length=5))
              True
          """
          return self.first_pos_in(other) is not None
  
      def factor_occurrences_in(self, other):
          r"""
          Return an iterator over all occurrences (including overlapping ones)
          of ``self`` in ``other`` in their order of appearance.
  
          EXAMPLES::
  
              sage: u = Word('121')
              sage: w = Word('121213211213')
              sage: list(u.factor_occurrences_in(w))
              [0, 2, 8]
          """
          if self.length() == 0:
              raise NotImplementedError("The factor must be non empty")
          p = self._pos_in(other, 0)
          while p is not None:
              yield p
              p = self._pos_in(other, p+1)
  
      def nb_factor_occurrences_in(self, other):
          r"""
          Return the number of times ``self`` appears as a factor
          in ``other``.
  
          EXAMPLES::
  
              sage: Word().nb_factor_occurrences_in(Word('123'))
              Traceback (most recent call last):
              ...
              NotImplementedError: The factor must be non empty
              sage: Word('123').nb_factor_occurrences_in(Word('112332312313112332121123'))
              4
              sage: Word('321').nb_factor_occurrences_in(Word('11233231231311233221123'))
              0
          """
          n = 0
          for _ in self.factor_occurrences_in(other):
              n += 1
          return n
  
      def nb_subword_occurrences_in(self, other):
          r"""
          Return the number of times ``self`` appears in ``other`` as a subword.
  
          This corresponds to the notion of `binomial coefficient` of two
          finite words whose properties are presented in the chapter of
          Lothaire's book written by Sakarovitch and Simon [1].
  
          INPUT:
  
          - ``other`` -- finite word
  
          EXAMPLES::
  
              sage: tm = words.ThueMorseWord()
  
              sage: u = Word([0,1,0,1])
              sage: u.nb_subword_occurrences_in(tm[:1000])
              2604124996
  
              sage: u = Word([0,1,0,1,1,0])
              sage: u.nb_subword_occurrences_in(tm[:100])
              20370432
  
          .. NOTE::
  
              This code, based on [2], actually compute the number of
              occurrences of all prefixes of ``self`` as subwords in all
              prefixes of ``other``.  In particular, its complexity is
              bounded by ``len(self) * len(other)``.
  
          TESTS::
  
              sage: Word('').nb_subword_occurrences_in(Word(''))
              1
              sage: parent(_)
              Integer Ring
              sage: v,u = Word(), Word('123')
              sage: v.nb_subword_occurrences_in(u)
              1
              sage: v,u = Word('123'), Word('1133432311132311112')
              sage: v.nb_subword_occurrences_in(u)
              11
              sage: v,u = Word('4321'), Word('1132231112233212342231112')
              sage: v.nb_subword_occurrences_in(u)
              0
              sage: v,u = Word('3'), Word('122332112321213')
              sage: v.nb_subword_occurrences_in(u)
              4
              sage: v,u = Word([]), words.ThueMorseWord()[:1000]
              sage: v.nb_subword_occurrences_in(u)
              1
  
          REFERENCES:
  
          - [1] M. Lothaire, Combinatorics on Words, Cambridge University
            Press, (1997).
          - [2] Mateescu, A., Salomaa, A., Salomaa, K. and Yu, S., A
            sharpening of the Parikh mapping. Theoret. Informatics Appl. 35
            (2001) 551-564.
          """
          # record the position of letters in self
          pos = defaultdict(list)
          for i,a in enumerate(self):
              pos[a].append(i)
          for a in pos:
              pos[a].reverse()
  
          # compute the occurrences of all prefixes of self as subwords in other
          occ = [ZZ.zero()] * (len(self)+1)
          occ[0] = ZZ.one()
          for a in other:
              for i in pos[a]:
                  occ[i+1] += occ[i]
  
          # return only the number of occurrences of self
          return occ[-1]
  
      def _return_words_list(self, fact):
          r"""
          Return the return words as a list in the order they appear in the word.
  
          INPUT:
  
          - ``fact`` -- a non-empty finite word
  
          OUTPUT:
  
          a Python list of finite words
  
          TESTS::
  
              sage: Word('baccabccbacbca')._return_words_list(Word('b'))
              [word: bacca, word: bcc, word: bac]
          """
          return list(self.return_words_iterator(fact))
  
      def return_words(self, fact):
          r"""
          Return the set of return words of ``fact`` in ``self``.
  
          This is the set of all factors starting by the given factor and ending
          just before the next occurrence of this factor. See [1] and [2].
  
          INPUT:
  
          - ``fact`` -- a non-empty finite word
  
          OUTPUT:
  
          a Python set of finite words
  
          EXAMPLES::
  
              sage: Word('21331233213231').return_words(Word('2'))
              {word: 213, word: 21331, word: 233}
              sage: Word().return_words(Word('213'))
              set()
              sage: Word('121212').return_words(Word('1212'))
              {word: 12}
  
          ::
  
              sage: TM = words.ThueMorseWord()[:1000]
              sage: sorted(TM.return_words(Word([0])))
              [word: 0, word: 01, word: 011]
  
          REFERENCES:
  
          -   [1] F. Durand, A characterization of substitutive sequences using
              return words, Discrete Math. 179 (1998) 89-101.
          -   [2] C. Holton, L.Q. Zamboni, Descendants of primitive substitutions,
              Theory Comput. Syst. 32 (1999) 133-157.
          """
          return set(self.return_words_iterator(fact))
  
      def complete_return_words(self, fact):
          r"""
          Return the set of complete return words of ``fact`` in ``self``.
  
          This is the set of all factors starting by the given factor and ending
          just after the next occurrence of this factor. See for instance [1].
  
          INPUT:
  
          - ``fact`` -- a non-empty finite word
  
          OUTPUT:
  
          a Python set of finite words
  
          EXAMPLES::
  
              sage: s = Word('21331233213231').complete_return_words(Word('2'))
              sage: sorted(s)
              [word: 2132, word: 213312, word: 2332]
              sage: Word('').complete_return_words(Word('213'))
              set()
              sage: Word('121212').complete_return_words(Word('1212'))
              {word: 121212}
  
          REFERENCES:
  
          -   [1] J. Justin, L. Vuillon, Return words in Sturmian and
              episturmian words, Theor. Inform. Appl. 34 (2000) 343--356.
          """
          return set(self.complete_return_words_iterator(fact))
  
      def return_words_derivate(self, fact):
          r"""
          Return the word generated by mapping a letter to each occurrence of
          the return words for the given factor dropping any dangling prefix and
          suffix. See for instance [1].
  
          EXAMPLES::
  
              sage: Word('12131221312313122').return_words_derivate(Word('1'))
              word: 123242
  
          REFERENCES:
  
          -   [1] F. Durand, A characterization of substitutive sequences using
              return words, Discrete Math. 179 (1998) 89--101.
          """
          idx = 0
          tab = {}
          ret = [tab.setdefault(w, len(tab)) + 1 for w in self._return_words_list(fact)]
          from sage.combinat.words.word import Word
          return Word(ret)
  
      def is_quasiperiodic(self):
          r"""
          Return ``True`` if ``self`` is quasiperiodic, and ``False`` otherwise.
  
          A finite or infinite word `w` is *quasiperiodic* if it can be
          constructed by concatenations and superpositions of one of its proper
          factors `u`, which is called a *quasiperiod* of `w`.
          See for instance [1], [2], and [3].
  
          EXAMPLES::
  
              sage: Word('abaababaabaababaaba').is_quasiperiodic()
              True
              sage: Word('abacaba').is_quasiperiodic()
              False
              sage: Word('a').is_quasiperiodic()
              False
              sage: Word().is_quasiperiodic()
              False
              sage: Word('abaaba').is_quasiperiodic()
              True
  
          REFERENCES:
  
          -   [1] A. Apostolico, A. Ehrenfeucht, Efficient detection of
              quasiperiodicities in strings, Theoret. Comput. Sci. 119 (1993)
              247--265.
          -   [2] S. Marcus, Quasiperiodic infinite words, Bull. Eur. Assoc.
              Theor. Comput. Sci. 82 (2004) 170-174.
          -   [3] A. Glen, F. Levé, G. Richomme, Quasiperiodic and Lyndon
              episturmian words, Preprint, 2008, :arxiv:`0805.0730`.
          """
          l = self.length()
          if l <= 1:
             return False
          for i in range(1, l - 1):
              return_lengths = [x.length() for x in self.return_words(self[:i])]
              if return_lengths != []:
                 if (max(return_lengths) <= i and self[l-i:l] == self[:i]):
                    return True
          return False
  
      def quasiperiods(self):
          r"""
          Return the quasiperiods of ``self`` as a list ordered from shortest to
          longest.
  
          Let `w` be a finite or infinite word. A *quasiperiod* of `w` is a
          proper factor `u` of `w` such that the occurrences of `u` in `w`
          entirely cover `w`, i.e., every position of `w` falls within some
          occurrence of `u` in `w`. See for instance [1], [2], and [3].
  
          EXAMPLES::
  
              sage: Word('abaababaabaababaaba').quasiperiods()
              [word: aba, word: abaaba, word: abaababaaba]
              sage: Word('abaaba').quasiperiods()
              [word: aba]
              sage: Word('abacaba').quasiperiods()
              []
  
          REFERENCES:
  
          -   [1] A. Apostolico, A. Ehrenfeucht, Efficient detection of
              quasiperiodicities in strings, Theoret. Comput. Sci. 119 (1993)
              247--265.
          -   [2] S. Marcus, Quasiperiodic infinite words, Bull. Eur. Assoc.
              Theor. Comput. Sci. 82 (2004) 170-174.
          -   [3] A. Glen, F. Levé, G. Richomme, Quasiperiodic and Lyndon
              episturmian words, Preprint, 2008, :arxiv:`0805.0730`.
          """
          l = self.length()
          if l <= 1:
             return []
          Q = []
          for i in range(1, l - 1):
              return_lengths = [x.length() for x in self.return_words(self[:i])]
              if return_lengths != []:
                 if (max(return_lengths) <= i and self[l-i:l] == self[:i]):
                    Q.append(self[:i])
          return Q
  
      def crochemore_factorization(self):
          r"""
          Return the Crochemore factorization of ``self`` as an ordered list of
          factors.
  
          The *Crochemore factorization* or the *Lempel-Ziv decomposition* of a
          finite word `w` is the unique factorization: `(x_1, x_2, \ldots, x_n)`
          of `w` with each `x_i` satisfying either: C1. `x_i` is a letter that
          does not appear in `u = x_1\ldots x_{i-1}`; C2. `x_i` is the longest
          prefix of `v = x_i\ldots x_n` that also has an occurrence beginning
          within `u = x_1\ldots x_{i-1}`. See [1].
  
          EXAMPLES::
  
              sage: x = Word('abababb')
              sage: x.crochemore_factorization()
              (a, b, abab, b)
              sage: mul(x.crochemore_factorization()) == x
              True
              sage: y = Word('abaababacabba')
              sage: y.crochemore_factorization()
              (a, b, a, aba, ba, c, ab, ba)
              sage: mul(y.crochemore_factorization()) == y
              True
              sage: x = Word([0,1,0,1,0,1,1])
              sage: x.crochemore_factorization()
              (0, 1, 0101, 1)
              sage: mul(x.crochemore_factorization()) == x
              True
  
          REFERENCES:
  
          -   [1] M. Crochemore, Recherche linéaire d'un carré dans un mot,
              C. R. Acad. Sci. Paris Sér. I Math. 296 (1983) 14 781--784.
          """
          T = self.implicit_suffix_tree()
          cuts = T.LZ_decomposition()
          c = Factorization([self[cuts[i]:cuts[i+1]] for i in range(len(cuts)-1)])
          return c
  
      LZ_decomposition = crochemore_factorization
  
      def evaluation_dict(self):
          r"""
          Return a dictionary keyed by the letters occurring in ``self`` with
          values the number of occurrences of the letter.
  
          EXAMPLES::
  
              sage: Word([2,1,4,2,3,4,2]).evaluation_dict()
              {1: 1, 2: 3, 3: 1, 4: 2}
              sage: Word('badbcdb').evaluation_dict()
              {'a': 1, 'b': 3, 'c': 1, 'd': 2}
              sage: Word().evaluation_dict()
              {}
  
          ::
  
              sage: f = Word('1213121').evaluation_dict() # keys appear in random order
              {'1': 4, '2': 2, '3': 1}
  
          TESTS::
  
              sage: f = Word('1213121').evaluation_dict()
              sage: f['1'] == 4
              True
              sage: f['2'] == 2
              True
              sage: f['3'] == 1
              True
          """
          return evaluation_dict(self)
  
      def evaluation_sparse(self):
          r"""
          Return a list representing the evaluation of ``self``. The entries of
          the list are two-element lists ``[a, n]``, where ``a`` is a letter
          occurring in ``self`` and ``n`` is the number of occurrences of ``a`` in ``self``.
  
          EXAMPLES::
  
              sage: Word([4,4,2,5,2,1,4,1]).evaluation_sparse()
              [(1, 2), (2, 2), (4, 3), (5, 1)]
              sage: Word("abcaccab").evaluation_sparse()
              [('a', 3), ('c', 3), ('b', 2)]
          """
          return self.evaluation_dict().items()
  
      def evaluation_partition(self):
          r"""
          Return the evaluation of the word w as a partition.
  
          EXAMPLES::
  
              sage: Word("acdabda").evaluation_partition()
              [3, 2, 1, 1]
              sage: Word([2,1,4,2,3,4,2]).evaluation_partition()
              [3, 2, 1, 1]
          """
          p = sorted(self.evaluation_dict().values(), reverse=True)
          from sage.combinat.partition import Partition
          if 0 in p:
              return Partition(p[:p.index(0)])
          else:
              return Partition(p)
  
      def overlap_partition(self, other, delay=0, p=None, involution=None) :
          r"""
          Return the partition of the alphabet induced by the overlap of
          ``self`` and ``other`` with the given ``delay``.
  
          The partition of the alphabet is given by the equivalence
          relation obtained from the symmetric, reflexive and transitive
          closure of the set of pairs of letters
          `R_{u,v,d} = \{ (u_k, v_{k-d}) : 0 \leq k < n, 0\leq k-d < m \}`
          where `u = u_0 u_1 \cdots u_{n-1}`, `v = v_0v_1\cdots v_{m-1}` are
          two words on the alphabet `A` and `d` is an integer.
  
          The equivalence relation defined by `R` is inspired from [1].
  
          INPUT:
  
          -  ``other`` -- word on the same alphabet as ``self``
          -  ``delay`` -- integer (default: ``0``)
          -  ``p`` -- disjoint sets data structure (optional, default: ``None``),
             a partition of the alphabet into disjoint sets to start with.
             If ``None``, each letter start in distinct equivalence classes.
          -  ``involution`` -- callable (optional, default: ``None``), an
             involution on the alphabet. If ``involution`` is not ``None``, the relation
             `R_{u,v,d} \cup R_{involution(u),involution(v),d}` is considered.
  
          OUTPUT:
  
          a disjoint set data structure
  
          EXAMPLES::
  
              sage: W = Words(list('abc') + list(range(6)))
              sage: u = W('abc')
              sage: v = W(range(5))
              sage: u.overlap_partition(v)
              {{0, 'a'}, {1, 'b'}, {2, 'c'}, {3}, {4}, {5}}
              sage: u.overlap_partition(v, 2)
              {{'a'}, {'b'}, {0, 'c'}, {1}, {2}, {3}, {4}, {5}}
              sage: u.overlap_partition(v, -1)
              {{0}, {1, 'a'}, {2, 'b'}, {3, 'c'}, {4}, {5}}
  
          You can re-use the same disjoint set and do more than one overlap::
  
              sage: p = u.overlap_partition(v, 2)
              sage: p
              {{'a'}, {'b'}, {0, 'c'}, {1}, {2}, {3}, {4}, {5}}
              sage: u.overlap_partition(v, 1, p)
              {{'a'}, {0, 1, 'b', 'c'}, {2}, {3}, {4}, {5}}
  
          The function  ``overlap_partition`` can be used to study equations
          on words. For example, if a word `w` overlaps itself with delay `d`, then
          `d` is a period of `w`::
  
              sage: W = Words(range(20))
              sage: w = W(range(14)); w
              word: 0,1,2,3,4,5,6,7,8,9,10,11,12,13
              sage: d = 5
              sage: p = w.overlap_partition(w, d)
              sage: m = WordMorphism(p.element_to_root_dict())
              sage: w2 = m(w); w2
              word: 56789567895678
              sage: w2.minimal_period() == d
              True
  
          If a word is equal to its reversal, then it is a palindrome::
  
              sage: W = Words(range(20))
              sage: w = W(range(17)); w
              word: 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16
              sage: p = w.overlap_partition(w.reversal(), 0)
              sage: m = WordMorphism(p.element_to_root_dict())
              sage: w2 = m(w); w2
              word: 01234567876543210
              sage: w2.parent()
              Finite words over {0, 1, 2, 3, 4, 5, 6, 7, 8, 17, 18, 19}
              sage: w2.is_palindrome()
              True
  
          If the reversal of a word `w` is factor of its square `w^2`, then
          `w` is symmetric, i.e. the product of two palindromes::
  
              sage: W = Words(range(10))
              sage: w = W(range(10)); w
              word: 0123456789
              sage: p = (w*w).overlap_partition(w.reversal(), 4)
              sage: m = WordMorphism(p.element_to_root_dict())
              sage: w2 = m(w); w2
              word: 0110456654
              sage: w2.is_symmetric()
              True
  
          If the image of the reversal of a word `w` under an involution `f`
          is factor of its square `w^2`, then `w` is `f`-symmetric::
  
              sage: W = Words([-11,-9,..,11])
              sage: w = W([1,3,..,11])
              sage: w
              word: 1,3,5,7,9,11
              sage: inv = lambda x:-x
              sage: f = WordMorphism(dict( (a, inv(a)) for a in W.alphabet()))
              sage: p = (w*w).overlap_partition(f(w).reversal(), 2, involution=f)
              sage: m = WordMorphism(p.element_to_root_dict())
              sage: m(w)
              word: 1,-1,5,7,-7,-5
              sage: m(w).is_symmetric(f)
              True
  
          TESTS::
  
              sage: W = Words('abcdef')
              sage: w = W('abc')
              sage: y = W('def')
              sage: w.overlap_partition(y, -3)
              {{'a'}, {'b'}, {'c'}, {'d'}, {'e'}, {'f'}}
              sage: w.overlap_partition(y, -2)
              {{'a', 'f'}, {'b'}, {'c'}, {'d'}, {'e'}}
              sage: w.overlap_partition(y, -1)
              {{'a', 'e'}, {'b', 'f'}, {'c'}, {'d'}}
              sage: w.overlap_partition(y, 0)
              {{'a', 'd'}, {'b', 'e'}, {'c', 'f'}}
              sage: w.overlap_partition(y, 1)
              {{'a'}, {'b', 'd'}, {'c', 'e'}, {'f'}}
              sage: w.overlap_partition(y, 2)
              {{'a'}, {'b'}, {'c', 'd'}, {'e'}, {'f'}}
              sage: w.overlap_partition(y, 3)
              {{'a'}, {'b'}, {'c'}, {'d'}, {'e'}, {'f'}}
              sage: w.overlap_partition(y, 4)
              {{'a'}, {'b'}, {'c'}, {'d'}, {'e'}, {'f'}}
  
          ::
  
              sage: W = Words(range(2))
              sage: w = W([0,1,0,1,0,1]); w
              word: 010101
              sage: w.overlap_partition(w, 0)
              {{0}, {1}}
              sage: w.overlap_partition(w, 1)
              {{0, 1}}
  
          ::
  
              sage: empty = Word()
              sage: empty.overlap_partition(empty, 'yo')
              Traceback (most recent call last):
              ...
              TypeError: delay (=yo) must be an integer
              sage: empty.overlap_partition(empty,2,'yo')
              Traceback (most recent call last):
              ...
              TypeError: p(=yo) is not a DisjointSet
  
          The ``involution`` input can be any callable::
  
              sage: w = Words([-5,..,5])([-5..5])
              sage: inv = lambda x:-x
              sage: w.overlap_partition(w, 2, involution=inv)
              {{-4, -2, 0, 2, 4}, {-5, -3, -1, 1, 3, 5}}
  
          REFERENCES:
  
          -   [1] S. Labbé, Propriétés combinatoires des `f`-palindromes,
              Mémoire de maîtrise en Mathématiques, Montréal, UQAM, 2008,
              109 pages.
          """
          if not isinstance(delay, (int, Integer)):
              raise TypeError("delay (=%s) must be an integer"%delay)
          elif delay < 0:
              return other.overlap_partition(self, -delay, p)
  
          from sage.sets.disjoint_set import DisjointSet_class
          if p is None:
              if self.parent().alphabet().cardinality() is Infinity:
                  raise ValueError("The alphabet of the parent must be finite")
              from sage.sets.disjoint_set import DisjointSet
              p = DisjointSet(self.parent().alphabet())
          elif not isinstance(p, DisjointSet_class):
              raise TypeError("p(=%s) is not a DisjointSet" % p)
  
          #Join the classes of each pair of letters that are one above the other
          from sage.combinat.words.morphism import WordMorphism
          S = zip(islice(self, delay, None), other)
          if involution is None:
              for (a,b) in S:
                  p.union(a, b)
          elif isinstance(involution, WordMorphism):
              for (a,b) in S:
                  p.union(a, b)
                  # take the first letter of the word
                  p.union(involution(a)[0], involution(b)[0])
          elif callable(involution):
              for (a,b) in S:
                  p.union(a, b)
                  p.union(involution(a), involution(b))
          else:
              raise TypeError("involution (=%s) must be callable"%involution)
          return p
  
      # TODO: requires a parent with a sortkey_letters method
      def standard_permutation(self):
          r"""
          Return the standard permutation of the word
          ``self`` on the ordered alphabet. It is defined as
          the permutation with exactly the same number of
          inversions as w. Equivalently, it is the permutation
          of minimal length whose inverse sorts ``self``.
  
          EXAMPLES::
  
              sage: w = Word([1,2,3,2,2,1]); w
              word: 123221
              sage: p = w.standard_permutation(); p
              [1, 3, 6, 4, 5, 2]
              sage: v = Word(p.inverse().action(w)); v
              word: 112223
              sage: filter(lambda q: q.length() <= p.length() and \
              ....:       q.inverse().action(w) == list(v), \
              ....:       Permutations(w.length()) )
              [[1, 3, 6, 4, 5, 2]]
  
          ::
  
              sage: w = Words([1,2,3])([1,2,3,2,2,1,2,1]); w
              word: 12322121
              sage: p = w.standard_permutation(); p
              [1, 4, 8, 5, 6, 2, 7, 3]
              sage: Word(p.inverse().action(w))
              word: 11122223
  
          ::
  
              sage: w = Words([3,2,1])([1,2,3,2,2,1,2,1]); w
              word: 12322121
              sage: p = w.standard_permutation(); p
              [6, 2, 1, 3, 4, 7, 5, 8]
              sage: Word(p.inverse().action(w))
              word: 32222111
  
          ::
  
              sage: w = Words('ab')('abbaba'); w
              word: abbaba
              sage: p = w.standard_permutation(); p
              [1, 4, 5, 2, 6, 3]
              sage: Word(p.inverse().action(w))
              word: aaabbb
  
          ::
  
              sage: w = Words('ba')('abbaba'); w
              word: abbaba
              sage: p = w.standard_permutation(); p
              [4, 1, 2, 5, 3, 6]
              sage: Word(p.inverse().action(w))
              word: bbbaaa
          """
          from sage.combinat.permutation import to_standard
          return to_standard(self, key=self.parent().sortkey_letters)
  
      def _s(self, i):
          r"""
          Implement Lascoux and Schützenberger `s_i` operator, swap the
          number of ``i`` and ``i+1`` in a word.
  
          EXAMPLES::
  
              sage: w = Word([1,1,2,1,2,2,1,3,1,2])
              sage: w._s(1)
              word: 1221221312
  
          TESTS::
  
              sage: w = Word([])
              sage: w._s(1)
              word:
              sage: w = Words(3)([2,1,2])
              sage: w._s(1).parent()
              Finite words over {1, 2, 3}
          """
          unpaired_i  = [] # positions of unpaired is
          unpaired_ip = [] # positions of unpaired i+1s
          for p,x in enumerate(self):
              if x == i:
                  if unpaired_ip:
                      unpaired_ip.pop()
                  else:
                      unpaired_i.append(p)
              elif x == i+1:
                  unpaired_ip.append(p)
  
          unpaired = unpaired_i + unpaired_ip
  
          out = list(self)
  
          # replace the unpaired subword i^a (i+1)^b
          # with i^b (i+1)^a
  
          for j,p in enumerate(unpaired):
              if j < len(unpaired_ip):
                  out[p] = i
              else:
                  out[p] = i+1
          return self.parent()(out, check=False)
  
      def _to_partition_content(self):
          r"""
          Return the conversion of ``self`` to a word with partition content using
          the `s_i` operators of Lascoux and Schützenberger.
  
          EXAMPLES::
  
              sage: w = Word([1,3,2,1,2,3,4,6,4,2,3,2])
              sage: w._to_partition_content()
              word: 132112454132
              sage: Word([])._to_partition_content()
              word:
          """
          if self.length() == 0:
              return self
  
          from sage.combinat.words.word import Word
          n = max(self)
          ev = Word( Words(n)(self).evaluation() )
          sig = ev.reversal().standard_permutation().reduced_word()
  
          # sig is now the reverse complement of a reduced word for a minimal
          # length permutation which would sort the letters of ev into a
          # partition
  
          out = self
          for i in reversed(sig):
              out = out._s(n-i)
          return out
  
      def cocharge(self):
          r"""
          Return the cocharge of ``self``.  For a word `w`, this can be defined as
          `n_{ev} - ch(w)`, where `ch(w)` is the charge of `w` and `ev` is the
          evaluation of `w`, and `n_{ev}` is `\sum_{i<j} min(ev_i, ev_j)`.
  
          EXAMPLES::
  
              sage: Word([1,2,3]).cocharge()
              0
              sage: Word([3,2,1]).cocharge()
              3
              sage: Word([1,1,2]).cocharge()
              0
              sage: Word([2,1,2]).cocharge()
              1
  
          TESTS::
  
              sage: Word([]).cocharge()
              0
          """
          return self.evaluation_partition().weighted_size() - self.charge()
  
      def charge(self, check=True):
          r"""
          Return the charge of ``self``.  This is defined as follows.
  
          If `w` is a permutation of length `n`, (in other words, the evaluation
          of `w` is `(1, 1, \dots, 1)`), the statistic charge(`w`) is given by
          `\sum_{i=1}^n c_i(w)` where `c_1(w) = 0` and `c_i(w)` is defined
          recursively by setting `p_i` equal to `1` if `i` appears to the right
          of `i-1` in `w` and `0` otherwise.  Then we set `c_i(w) = c_{i-1}(w) +
          p_i`.
  
          EXAMPLES::
  
              sage: Word([1, 2, 3]).charge()
              3
              sage: Word([3, 5, 1, 4, 2]).charge() == 0 + 1 + 1 + 2 + 2
              True
  
          If `w` is not a permutation, but the evaluation of `w` is a partition,
          the charge of `w` is defined to be the sum of its charge subwords
          (each of which will be a permutation).  The first charge subword is
          found by starting at the end of `w` and moving left until the first
          `1` is found.  This is marked, and we continue to move to the left
          until the first `2` is found, wrapping around from the beginning of
          the word back to the end, if necessary.  We mark this `2`, and
          continue on until we have marked the largest letter in `w`.  The
          marked letters, with relative order preserved, form the first charge
          subword of `w`.  This subword is removed, and the next charge subword
          is found in the same manner from the remaining letters.  In the
          following example, `w1, w2, w3` are the charge subwords of `w`.
  
          EXAMPLES::
  
              sage: w = Word([5,2,3,4,4,1,1,1,2,2,3])
              sage: w1 = Word([5, 2, 4, 1, 3])
              sage: w2 = Word([3, 4, 1, 2])
              sage: w3 = Word([1, 2])
              sage: w.charge() == w1.charge() + w2.charge() + w3.charge()
              True
  
          Finally, if `w` does not have partition content, we apply the
          Lascoux-Schützenberger standardization operators `s_i` in such a
          manner as to obtain a word with partition content. (The word we obtain
          is independent of the choice of operators.)  The charge is then
          defined to be the charge of this word::
  
              sage: Word([3,3,2,1,1]).charge()
              0
              sage: Word([1,2,3,1,2]).charge()
              2
  
          Note that this differs from the definition of charge given in
          Macdonald's book.  The difference amounts to a choice of
          reading a word from left-to-right or right-to-left.  The choice in
          Sage was made to agree with the definition of a reading word of a
          tableau in Sage, and seems to be the more common convention in the
          literature.
  
          REFERENCES:
  
          [1] Ian Macdonald, *Symmetric Functions and Hall Polynomials* second
          edition, 1995, Oxford University Press
  
          [2] A. Lascoux, L. Lapointe, and J. Morse.  *Tableau atoms and a new
          Macdonald positivity conjecture.* Duke Math Journal, **116 (1)**,
          2003.  :arxiv:`math/0008073`
  
          [3] A. Lascoux, B. Leclerc, and J.Y. Thibon.  *The Plactic Monoid*.
          Survey article available at
          [http://www-igm.univ-mlv.fr/~jyt/ARTICLES/plactic.ps]
  
          TESTS::
  
              sage: Word([1,1,2,2,3]).charge()
              4
              sage: Word([3,1,1,2,2]).charge()
              3
              sage: Word([2,1,1,2,3]).charge()
              2
              sage: Word([2,1,1,3,2]).charge()
              2
              sage: Word([3,2,1,1,2]).charge()
              1
              sage: Word([2,2,1,1,3]).charge()
              1
              sage: Word([3,2,2,1,1]).charge()
              0
              sage: Word([]).charge()
              0
          """
          if check:
              ev_dict = self.evaluation_dict()
              ordered_alphabet = sorted(ev_dict,
                                        key=self.parent().sortkey_letters)
              evaluation = [ev_dict[a] for a in ordered_alphabet]
              from sage.combinat.partition import Partitions
              if evaluation not in Partitions():
                  return self._to_partition_content().charge()
          res = 0
          w = self.to_integer_list()
          while len(w) != 0:
              i =len(w) - 1
              l = min(w)
              index = 0
              while len(w) != 0 and l <= max(w):
                  while w[i] != l:
                      i -= 1
                      if i < 0:
                          i = len(w) - 1
                          index += 1
                  res += index
                  l += 1
                  w.pop(i)
                  i -= 1
                  if i < 0:
                      i = len(w) - 1
                      index += 1
          return res
  
      def BWT(self):
          r"""
          Return the Burrows-Wheeler Transform (BWT) of ``self``.
  
          The *Burrows-Wheeler transform* of a finite word `w` is obtained
          from `w` by first listing the conjugates of `w` in lexicographic order
          and then concatenating the final letters of the conjugates in this
          order. See [1].
  
          EXAMPLES::
  
              sage: Word('abaccaaba').BWT()
              word: cbaabaaca
              sage: Word('abaab').BWT()
              word: bbaaa
              sage: Word('bbabbaca').BWT()
              word: cbbbbaaa
              sage: Word('aabaab').BWT()
              word: bbaaaa
              sage: Word().BWT()
              word:
              sage: Word('a').BWT()
              word: a
  
          REFERENCES:
  
          -   [1] M. Burrows, D.J. Wheeler, "A block-sorting lossless data
              compression algorithm", HP Lab Technical Report, 1994, available
              at http://www.hpl.hp.com/techreports/Compaq-DEC/SRC-RR-124.html
          """
          if self.is_empty():
             return self
          conjugates = sorted(self._conjugates_list())
          return self.parent()([x[x.length()-1] for x in conjugates], check=False)
  
      def iterated_left_palindromic_closure(self, f=None):
          r"""
          Return the iterated left (``f``-)palindromic closure of ``self``.
  
          INPUT:
  
          - ``f`` -- involution (default: ``None``) on the alphabet of ``self``.
            It must be callable on letters as well as words (e.g. ``WordMorphism``).
  
          OUTPUT:
  
          word -- the left iterated ``f``-palindromic closure of ``self``.
  
          EXAMPLES::
  
              sage: Word('123').iterated_left_palindromic_closure()
              word: 3231323
              sage: f = WordMorphism('a->b,b->a')
              sage: Word('ab').iterated_left_palindromic_closure(f=f)
              word: abbaab
              sage: Word('aab').iterated_left_palindromic_closure(f=f)
              word: abbaabbaab
  
          TESTS:
  
          If ``f`` is not an involution::
  
              sage: f = WordMorphism('a->b,b->b')
              sage: Word('aab').iterated_left_palindromic_closure(f=f)
              Traceback (most recent call last):
              ...
              TypeError: self (=a->b, b->b) is not an endomorphism
  
          REFERENCES:
  
          -   A. de Luca, A. De Luca, Pseudopalindrome closure operators
              in free monoids, Theoret. Comput. Sci. 362 (2006) 282--300.
          """
          if f is None:
              return self.reversal().iterated_right_palindromic_closure(f=f)
          else:
              from sage.combinat.words.morphism import WordMorphism
              f = WordMorphism(f)
              return f(self).reversal().iterated_right_palindromic_closure(f=f)
  
      def count(self, letter):
          r"""
          Count the number of occurrences of ``letter`` in ``self``.
  
          EXAMPLES::
  
              sage: Word('abbabaab').count('a')
              4
          """
          return Integer(sum(1 for a in self if a == letter))
  
      def balance(self):
          r"""
          Return the balance of ``self``.
  
          The balance of a word is the smallest number `q` such that ``self`` is
          `q`-balanced [1].
  
          A finite or infinite word `w` is said to be `q`-*balanced* if for
          any two factors `u`, `v` of `w` of the same length, the difference
          between the number of `x`'s in each of `u` and `v` is at most `q`
          for all letters `x` in the alphabet of `w`. A `1`-balanced word is
          simply said to be balanced. See Chapter 2 of [2].
  
          OUTPUT:
  
          integer
  
          EXAMPLES::
  
              sage: Word('1111111').balance()
              0
              sage: Word('001010101011').balance()
              2
              sage: Word('0101010101').balance()
              1
  
          ::
  
              sage: w = Word('11112222')
              sage: w.is_balanced(2)
              False
              sage: w.is_balanced(3)
              False
              sage: w.is_balanced(4)
              True
              sage: w.is_balanced(5)
              True
              sage: w.balance()
              4
  
          TESTS::
  
              sage: Word('1111122222').balance()
              5
              sage: Word('').balance()
              0
              sage: Word('1').balance()
              0
              sage: Word('12').balance()
              1
              sage: Word('1112').balance()
              1
  
          REFERENCES:
  
          -  [1] I. Fagnot, L. Vuillon, Generalized balances in Sturmian words,
             Discrete Applied Mathematics 121 (2002), 83--101.
          -  [2] M. Lothaire, Algebraic Combinatorics On Words, vol. 90 of
             Encyclopedia of Mathematics and its Applications, Cambridge
             University Press, U.K., 2002.
          """
          alphabet = self.letters()
          best = 0
          for i in range(1, self.length()):
              start = iter(self)
              end = iter(self)
              abelian = dict(zip(alphabet, [0]*len(alphabet)))
              for _ in range(i):
                  abelian[next(end)] += 1
              abel_max = abelian.copy()
              abel_min = abelian.copy()
              for _ in range(self.length() - i):
                  lost = next(start)
                  gain = next(end)
                  abelian[gain] += 1
                  abelian[lost] -= 1
                  abel_max[gain] = max(abel_max[gain], abelian[gain])
                  abel_min[lost] = min(abel_min[lost], abelian[lost])
              best = max(best, max(abel_max[a] - abel_min[a] for a in alphabet))
          return best
  
      def is_balanced(self, q=1):
          r"""
          Return ``True`` if ``self`` is ``q``-balanced, and ``False`` otherwise.
  
          A finite or infinite word `w` is said to be `q`-*balanced* if for
          any two factors `u`, `v` of `w` of the same length, the difference
          between the number of `x`'s in each of `u` and `v` is at most `q`
          for all letters `x` in the alphabet of `w`. A `1`-balanced word is
          simply said to be balanced. See for instance [1] and Chapter 2 of
          [2].
  
          INPUT:
  
          - ``q`` -- integer (default: ``1``), the balance level
  
          OUTPUT:
  
          boolean -- the result
  
          EXAMPLES::
  
              sage: Word('1213121').is_balanced()
              True
              sage: Word('1122').is_balanced()
              False
              sage: Word('121333121').is_balanced()
              False
              sage: Word('121333121').is_balanced(2)
              False
              sage: Word('121333121').is_balanced(3)
              True
              sage: Word('121122121').is_balanced()
              False
              sage: Word('121122121').is_balanced(2)
              True
  
          TESTS::
  
              sage: Word('121122121').is_balanced(-1)
              Traceback (most recent call last):
              ...
              TypeError: the balance level must be a positive integer
              sage: Word('121122121').is_balanced(0)
              Traceback (most recent call last):
              ...
              TypeError: the balance level must be a positive integer
              sage: Word('121122121').is_balanced('a')
              Traceback (most recent call last):
              ...
              TypeError: the balance level must be a positive integer
  
          REFERENCES:
  
          -   [1] J. Cassaigne, S. Ferenczi, L.Q. Zamboni, Imbalances in
              Arnoux-Rauzy sequences, Ann. Inst. Fourier (Grenoble) 50 (2000)
              1265--1276.
          -   [2] M. Lothaire, Algebraic Combinatorics On Words, vol. 90 of
              Encyclopedia of Mathematics and its Applications, Cambridge
              University Press, U.K., 2002.
          """
          if not isinstance(q, (int, Integer)) or q <= 0:
              raise TypeError("the balance level must be a positive integer")
          alphabet = self.letters()
          for i in range(2, self.length()):
              empty_sets = [set() for _ in range(len(alphabet))]
              tab = dict(zip(alphabet, empty_sets))
              for fact in self.factor_iterator(i):
                  evaluation_dict = fact.evaluation_dict()
                  for a in alphabet:
                      tab[a].add(evaluation_dict.get(a, 0))
              for t in tab.values():
                  if len(t) > q+1:
                      return False
          return True
  
      def abelian_vectors(self, n):
          r"""
          Return the abelian vectors of factors of length ``n`` of ``self``.
  
          The vectors are defined w.r.t. the order of the alphabet of the
          parent.
  
          OUTPUT:
  
          a set of tuples
  
          EXAMPLES::
  
              sage: W = Words([0,1,2])
              sage: w = W([0,1,1,0,1,2,0,2,0,2])
              sage: w.abelian_vectors(3)
              {(1, 0, 2), (1, 1, 1), (1, 2, 0), (2, 0, 1)}
              sage: w[:5].abelian_vectors(3)
              {(1, 2, 0)}
              sage: w[5:].abelian_vectors(3)
              {(1, 0, 2), (2, 0, 1)}
  
          ::
  
              sage: w = words.FibonacciWord()[:100]
              sage: sorted(w.abelian_vectors(0))
              [(0, 0)]
              sage: sorted(w.abelian_vectors(1))
              [(0, 1), (1, 0)]
              sage: sorted(w.abelian_vectors(7))
              [(4, 3), (5, 2)]
  
          The word must be defined with a parent on a finite alphabet::
  
              sage: from itertools import count
              sage: w = Word(count(), alphabet=NN)
              sage: w[:2].abelian_vectors(2)
              Traceback (most recent call last):
              ...
              TypeError: The alphabet of the parent is infinite; define the
              word with a parent on a finite alphabet
  
          TESTS::
  
              sage: W = Words([0, 1])
              sage: w = W([0,0,0])
              sage: sorted(w.abelian_vectors(3))
              [(3, 0)]
              sage: w = W([0,0,0,1])
              sage: sorted(w.abelian_vectors(3))
              [(2, 1), (3, 0)]
              sage: w = W([0,0,0,1,1])
              sage: sorted(w.abelian_vectors(3))
              [(1, 2), (2, 1), (3, 0)]
              sage: w = W([0,0,0,1,1,1])
              sage: sorted(w.abelian_vectors(3))
              [(0, 3), (1, 2), (2, 1), (3, 0)]
  
          ::
  
              sage: w = Word([0,1,0], alphabet=[0,1])
              sage: w.abelian_complexity(3)
              1
              sage: w.abelian_complexity(4)
              0
  
          """
          alphabet = self.parent().alphabet()
          size = alphabet.cardinality()
          if size == float('inf'):
              raise TypeError("The alphabet of the parent is infinite; define"
                     " the word with a parent on a finite alphabet")
          S = set()
          if n > self.length():
              return S
          rank = dict((letter,i) for i,letter in enumerate(alphabet))
          start = iter(self)
          end = iter(self)
          abelian = [0] * size
          for _ in range(n):
              abelian[rank[next(end)]] += 1
          S.add(tuple(abelian))
          for letter in end:
              abelian[rank[letter]] += 1
              abelian[rank[next(start)]] -= 1
              S.add(tuple(abelian))
          return S
  
      def abelian_complexity(self, n):
          r"""
          Return the number of abelian vectors of factors of length ``n`` of ``self``.
  
          EXAMPLES::
  
              sage: w = words.FibonacciWord()[:100]
              sage: [w.abelian_complexity(i) for i in range(20)]
              [1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]
  
          ::
  
              sage: w = words.ThueMorseWord()[:100]
              sage: [w.abelian_complexity(i) for i in range(20)]
              [1, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2]
  
          """
          return len(self.abelian_vectors(n))
  
      def sturmian_desubstitute_as_possible(self):
          r"""
          Sturmian-desubstitute the word ``self`` as much as possible.
  
          The finite word ``self`` must be defined on a two-letter
          alphabet or use at most two letters.
  
          It can be Sturmian desubstituted if one letter appears
          isolated: the Sturmian desubstitution consists in removing one
          letter per run of the non-isolated letter. The accelerated
          Sturmian desubstitution consists in removing a run equal to
          the length of the shortest inner run from any run of the
          non-isolated letter (including possible leading and trailing
          runs even if they have shorter length). The (accelerated)
          Sturmian desubstitution is done as much as possible. A word is
          a factor of a Sturmian word if, and only if, the result is the
          empty word.
  
          OUTPUT:
  
          a finite word defined on a two-letter alphabet
  
          EXAMPLES::
  
              sage: u = Word('10111101101110111',alphabet='01') ; u
              word: 10111101101110111
              sage: v = u.sturmian_desubstitute_as_possible() ; v
              word: 01100101
              sage: v == v.sturmian_desubstitute_as_possible()
              True
  
              sage: Word('azaazaaazaaazaazaaaz', alphabet='az').sturmian_desubstitute_as_possible()
              word:
  
  
          TESTS::
  
              sage: w = Word('azazaza', alphabet='aze')
              sage: w.sturmian_desubstitute_as_possible()
              word:
              sage: Word('aze').sturmian_desubstitute_as_possible()
              Traceback (most recent call last):
              ...
              TypeError: your word must be defined on a binary alphabet or use at most two different letters
              sage: Word('azaaazaazaazaaazaaza', alphabet='az').sturmian_desubstitute_as_possible()
              word:
              sage: Word('azaaazaazaazaaazaaaza', alphabet='az').sturmian_desubstitute_as_possible()
              word: azzaa
  
          Boundary effects::
  
              sage: Word('', alphabet='az').sturmian_desubstitute_as_possible()
              word:
              sage: Word('azzzzz', alphabet='az').sturmian_desubstitute_as_possible()
              word:
              sage: Word('zzzzza', alphabet='az').sturmian_desubstitute_as_possible()
              word:
              sage: Word('aaaaazaaaaaaaaa', alphabet='az').sturmian_desubstitute_as_possible()
              word:
              sage: Word('aaaaaaaaaaaaaa', alphabet='az').sturmian_desubstitute_as_possible()
              word:
  
          Boundary effects without alphabet::
  
              sage: Word('').sturmian_desubstitute_as_possible()
              word:
              sage: Word('azzzzz').sturmian_desubstitute_as_possible()
              word:
              sage: Word('zzzzza').sturmian_desubstitute_as_possible()
              word:
              sage: Word('aaaaazaaaaaaaaa').sturmian_desubstitute_as_possible()
              word:
              sage: Word('aaaaaaaaaaaaaa').sturmian_desubstitute_as_possible()
              word:
  
          Idempotence::
  
              sage: r = words.RandomWord(randint(1,15)).sturmian_desubstitute_as_possible() ; r == r.sturmian_desubstitute_as_possible()
              True
  
          AUTHOR:
  
          -   Thierry Monteil
          """
          if self.is_empty():
              return self
          W = self.parent()
          if W.alphabet().cardinality()== 2:
              alphabet = W.alphabet()
          else:
              alphabet = self.letters()
              if len(alphabet) > 2:
                  raise TypeError('your word must be defined on a binary alphabet or use at most two different letters')
              elif len(alphabet) < 2:
                  return W()
          word_from_letter = {l:W([l],datatype="list",check=False) for l in alphabet}
          is_prefix = True
          current_run_length = 0
          prefix_length = 0
          prefix_letter = self[0]
          is_isolated = {alphabet[0]:True,alphabet[1]:True}
          minimal_run = {alphabet[0]:Infinity,alphabet[1]:Infinity}
          maximal_run = {alphabet[0]:0,alphabet[1]:0}
          runs = {alphabet[0]:[],alphabet[1]:[]}
          for i in self:
              if is_prefix:
                  if i == prefix_letter:
                      prefix_length += 1
                      if prefix_length > 1:
                          is_isolated[i] = False
                  else:
                      is_prefix = False
                      current_run_length = 1
                      previous_letter = i
              else:
                  if i == previous_letter:
                      current_run_length += 1
                      if current_run_length > 1:
                          is_isolated[i] = False
                  else:
                      runs[previous_letter].append(current_run_length)
                      minimal_run[previous_letter] = min(minimal_run[previous_letter],current_run_length)
                      maximal_run[previous_letter] = max(maximal_run[previous_letter],current_run_length)
                      current_run_length = 1
                      previous_letter = i
          # at this point, previous_letter is the suffix letter and current_run_length is the suffix length
          if not (is_isolated[alphabet[0]] or is_isolated[alphabet[1]]):
              return self
          elif is_isolated[alphabet[0]] and is_isolated[alphabet[1]]:
              return W()
          else:
              if is_isolated[alphabet[0]]:
                  l_isolated = alphabet[0] #the isolated letter
                  l_running = alphabet[1] #the running letter (non-isolated)
              else:
                  l_isolated = alphabet[1]
                  l_running = alphabet[0]
              w_isolated = word_from_letter[l_isolated] #the word associated to the isolated letter
              w_running = word_from_letter[l_running] #the word associated to the running letter
              min_run = minimal_run[l_running]
              if (prefix_letter == l_isolated) or (prefix_length <= min_run):
                  desubstitued_word = W()
              else:
                  desubstitued_word = w_running ** (prefix_length - min_run)
              for i in runs[l_running]:
                  desubstitued_word = desubstitued_word + w_isolated + w_running ** (i - min_run)
              if (current_run_length > 0):
                  desubstitued_word = desubstitued_word + w_isolated
                  if (previous_letter == l_running) and (current_run_length > min_run):
                      desubstitued_word = desubstitued_word + w_running ** (current_run_length - min_run)
              return desubstitued_word.sturmian_desubstitute_as_possible()
  
  
      def is_sturmian_factor(self):
          r"""
          Tell whether ``self`` is a factor of a Sturmian word.
  
          The finite word ``self`` must be defined on a two-letter alphabet.
  
          Equivalently, tells whether ``self`` is balanced. The
          advantage over the ``is_balanced`` method is that this one runs in
          linear time whereas ``is_balanced`` runs in quadratic time.
  
          OUTPUT:
  
          boolean -- the result
  
          EXAMPLES::
  
              sage: w = Word('0111011011011101101',alphabet='01')
              sage: w.is_sturmian_factor()
              True
  
          ::
  
              sage: words.LowerMechanicalWord(random(),alphabet='01')[:100].is_sturmian_factor()
              True
              sage: words.CharacteristicSturmianWord(random())[:100].is_sturmian_factor()
              True
  
          ::
  
              sage: w = Word('aabb',alphabet='ab')
              sage: w.is_sturmian_factor()
              False
  
              sage: s1 = WordMorphism('a->ab,b->b')
              sage: s2 = WordMorphism('a->ba,b->b')
              sage: s3 = WordMorphism('a->a,b->ba')
              sage: s4 = WordMorphism('a->a,b->ab')
              sage: W = Words('ab')
              sage: w = W('ab')
              sage: for i in range(8): w = choice([s1,s2,s3,s4])(w)
              sage: w
              word: abaaabaaabaabaaabaaabaabaaabaabaaabaaaba...
              sage: w.is_sturmian_factor()
              True
  
          Famous words::
  
              sage: words.FibonacciWord()[:100].is_sturmian_factor()
              True
              sage: words.ThueMorseWord()[:1000].is_sturmian_factor()
              False
              sage: words.KolakoskiWord()[:1000].is_sturmian_factor()
              False
  
          REFERENCES:
  
          .. [Arn2002] \P. Arnoux, Sturmian sequences, in Substitutions in Dynamics,
             N. Pytheas Fogg (Ed.), Arithmetics, and Combinatorics (Lecture
             Notes in Mathematics, Vol. 1794), 2002.
          .. [Ser1985] \C. Series. The geometry of Markoff numbers. The Mathematical
             Intelligencer, 7(3):20--29, 1985.
          .. [SU2009] \J. Smillie and C. Ulcigrai. Symbolic coding for linear
             trajectories in the regular octagon, :arxiv:`0905.0871`, 2009.
  
          AUTHOR:
  
          -   Thierry Monteil
          """
          return self.sturmian_desubstitute_as_possible().is_empty()
  
  
      def is_tangent(self):
          r"""
          Tell whether ``self`` is a tangent word.
  
          The finite word ``self`` must be defined on a two-letter alphabet.
  
          A binary word is said to be *tangent* if it can appear in
          infinitely many cutting sequences of a smooth curve, where each
          cutting sequence is observed on a progressively smaller grid.
  
          This class of words strictly contains the class of `1`-balanced
          words, and is strictly contained in the class of `2`-balanced words.
  
          This method runs in linear time.
  
          OUTPUT:
  
          boolean -- the result
  
          EXAMPLES::
  
              sage: w = Word('01110110110111011101',alphabet='01')
              sage: w.is_tangent()
              True
  
          Some tangent words may not be balanced::
  
              sage: Word('aabb',alphabet='ab').is_balanced()
              False
              sage: Word('aabb',alphabet='ab').is_tangent()
              True
  
          Some `2`-balanced words may not be tangent::
  
              sage: Word('aaabb',alphabet='ab').is_tangent()
              False
              sage: Word('aaabb',alphabet='ab').is_balanced(2)
              True
  
          Famous words::
  
              sage: words.FibonacciWord()[:100].is_tangent()
              True
              sage: words.ThueMorseWord()[:1000].is_tangent()
              True
              sage: words.KolakoskiWord()[:1000].is_tangent()
              False
  
          REFERENCES:
  
          .. [Mon2010] \T. Monteil, The asymptotic language of smooth curves, talk
             at LaCIM2010.
  
          AUTHOR:
  
          -   Thierry Monteil
          """
          if (self.parent().alphabet().cardinality() != 2):
              raise TypeError('your word must be defined on a binary alphabet')
          [a,b] = self.parent().alphabet()
          mini = 0
          maxi = 0
          height = 0
          for i in self.sturmian_desubstitute_as_possible():
              if i == a:
                  height = height + 1
                  maxi = max(maxi , height)
              if i == b:
                  height = height - 1
                  mini = min(mini , height)
          return (maxi - mini <= 2)
  
  
      # TODO.
      # 1. Those three swap functions should use the cmp of python.
      # 2. The actual code should then be copied as is in the Word_over_Alphabet
      # and continue to use the parent cmp
      # 3. Once Word can define Words over alphabet, the examples
      # should be updated appropriately.
  
      def swap(self, i, j=None):
          r"""
          Return the word `w` with entries at positions ``i`` and
          ``j`` swapped. By default, ``j = i+1``.
  
          EXAMPLES::
  
              sage: Word([1,2,3]).swap(0,2)
              word: 321
              sage: Word([1,2,3]).swap(1)
              word: 132
              sage: Word("abba").swap(1,-1)
              word: aabb
          """
          if j is None:
              j = i+1
          new = list(self)
          (new[i], new[j]) = (new[j], new[i])
          from sage.combinat.words.word import Word
          return Word(new)
  
      def swap_increase(self, i):
          r"""
          Return the word with positions ``i`` and ``i+1`` exchanged
          if ``self[i] > self[i+1]``. Otherwise, it returns ``self``.
  
          EXAMPLES::
  
              sage: w = Word([1,3,2])
              sage: w.swap_increase(1)
              word: 123
              sage: w.swap_increase(0)
              word: 132
              sage: w.swap_increase(0) is w
              True
              sage: Words("ab")("abba").swap_increase(0)
              word: abba
              sage: Words("ba")("abba").swap_increase(0)
              word: baba
          """
          key = self._parent.sortkey_letters
          if key(self[i]) > key(self[i + 1]):
              return self.swap(i)
          else:
              return self
  
      def swap_decrease(self, i):
          r"""
          Return the word with positions ``i`` and ``i+1`` exchanged
          if ``self[i] < self[i+1]``. Otherwise, it returns ``self``.
  
          EXAMPLES::
  
              sage: w = Word([1,3,2])
              sage: w.swap_decrease(0)
              word: 312
              sage: w.swap_decrease(1)
              word: 132
              sage: w.swap_decrease(1) is w
              True
              sage: Words("ab")("abba").swap_decrease(0)
              word: baba
              sage: Words("ba")("abba").swap_decrease(0)
              word: abba
          """
          key = self._parent.sortkey_letters
          if key(self[i]) < key(self[i + 1]):
              return self.swap(i)
          else:
              return self
  
      def abelian_vector(self, alphabet=None):
          r"""
          Return the abelian vector of ``self`` counting the occurrences of each letter.
  
          The vector is defined w.r.t. the order of the alphabet of the
          parent. See also :meth:`evaluation_dict`.
  
          INPUT:
  
          - ``self`` -- word having a parent on a finite alphabet
          - ``alphabet`` -- *DEPRECATED*
  
          OUTPUT:
  
          a list
  
          EXAMPLES::
  
              sage: W = Words('ab')
              sage: W('aaabbbbb').abelian_vector()
              [3, 5]
              sage: W('a').abelian_vector()
              [1, 0]
              sage: W().abelian_vector()
              [0, 0]
  
          The argument ``alphabet`` is deprecated::
  
              sage: Word('aabaa').abelian_vector('abc')
              doctest:...: DeprecationWarning: The argument alphabet of
              methods abelian_vector and parikh_vector is deprecated and will
              be removed in a future version of Sage. In order to fix this,
              you must define your word on a parent with a finite alphabet.
              See http://trac.sagemath.org/17058 for details.
              [4, 1, 0]
  
          You may fix the above deprecated use of the ``alphabet`` argument this way::
  
              sage: W = Words('abc')
              sage: W('aabaa').abelian_vector()
              [4, 1, 0]
  
          TESTS::
  
              sage: W = Words()
              sage: W('aabaa').abelian_vector()
              Traceback (most recent call last):
              ...
              TypeError: The alphabet of the parent is infinite; define the
              word with a parent on a finite alphabet or use
              evaluation_dict() instead
          """
          if alphabet is None:
              if self.parent().alphabet().cardinality() is Infinity:
                  raise TypeError("The alphabet of the parent is infinite; define "
                          "the word with a parent on a finite alphabet or use "
                          "evaluation_dict() instead")
              alphabet = self.parent().alphabet()
          else:
              from sage.misc.superseded import deprecation
              deprecation(17058, "The argument alphabet of methods abelian_vector "
                          "and parikh_vector is deprecated and will be "
                          "removed in a future version of Sage. In order to "
                          "fix this, you must define your word on a parent "
                          "with a finite alphabet.")
  
          ev_dict = self.evaluation_dict()
          return [ev_dict.get(a,0) for a in alphabet]
  
      parikh_vector = deprecated_function_alias(17058, abelian_vector)
      evaluation = abelian_vector
  
      def robinson_schensted(self):
          """
          Return the semistandard tableau and standard tableau pair
          obtained by running the Robinson-Schensted algorithm on ``self``.
  
          This can also be done by running
          :func:`~sage.combinat.rsk.RSK` on ``self``.
  
          EXAMPLES::
  
              sage: Word([1,1,3,1,2,3,1]).robinson_schensted()
              [[[1, 1, 1, 1, 3], [2], [3]], [[1, 2, 3, 5, 6], [4], [7]]]
          """
          from sage.combinat.rsk import RSK
          return RSK(self)
  
      def _rsk_iter(self):
          r"""
          Return an iterator for :func:`~sage.combinat.rsk.RSK`.
  
          Yields pairs `(i, w_i)` for a word `w = w_1 w_2 \cdots w_k`.
  
          EXAMPLES::
  
              sage: for x in Word([1,1,3,1,2,3,1])._rsk_iter(): x
              ...
              (1, 1)
              (2, 1)
              (3, 3)
              (4, 1)
              (5, 2)
              (6, 3)
              (7, 1)
          """
          return zip(range(1, len(self) + 1), self)
  
      def shuffle(self, other, overlap=0):
          r"""
          Return the combinatorial class representing the shuffle product
          between words ``self`` and ``other``. This consists of all words of length
          ``self.length()+other.length()`` that have both ``self`` and ``other`` as
          subwords.
  
          If ``overlap`` is non-zero, then the combinatorial class representing
          the shuffle product with overlaps is returned. The calculation of
          the shift in each overlap is done relative to the order of the
          alphabet. For example, `a` shifted by `a` is `b` in the alphabet
          `[a, b, c]` and `0` shifted by `1` in `[0, 1, 2, 3]` is `2`.
  
          INPUT:
  
          -  ``other`` -- finite word
          -  ``overlap`` -- (default: ``0``) integer or ``True``
  
          OUTPUT:
  
          combinatorial class of shuffle product of ``self`` and ``other``
  
          EXAMPLES::
  
              sage: ab = Word("ab")
              sage: cd = Word("cd")
              sage: sp = ab.shuffle(cd); sp
              Shuffle product of word: ab and word: cd
              sage: sp.cardinality()
              6
              sage: sp.list()
              [word: abcd, word: acbd, word: acdb, word: cabd, word: cadb, word: cdab]
              sage: w = Word([0,1])
              sage: u = Word([2,3])
              sage: w.shuffle(w)
              Shuffle product of word: 01 and word: 01
              sage: u.shuffle(u)
              Shuffle product of word: 23 and word: 23
              sage: w.shuffle(u)
              Shuffle product of word: 01 and word: 23
              sage: w.shuffle(u,2)
              Overlapping shuffle product of word: 01 and word: 23 with 2 overlaps
          """
          if overlap == 0:
              from sage.combinat.words.shuffle_product import ShuffleProduct_w1w2
              return ShuffleProduct_w1w2(self, other)
          else:
              if any(a not in ZZ for a in self) or any(a not in ZZ for a in other):
                  raise ValueError("for a nonzero overlap, words must contain integers as letters")
              if overlap is True:
                  from sage.combinat.words.shuffle_product import ShuffleProduct_overlapping
                  return ShuffleProduct_overlapping(self, other)
              elif isinstance(overlap, (int,Integer)):
                  from sage.combinat.words.shuffle_product import ShuffleProduct_overlapping_r
                  return ShuffleProduct_overlapping_r(self, other, overlap)
              raise ValueError('overlapping must be True or an integer')
  
      def shifted_shuffle(self, other, shift=None):
          r"""
          Return the combinatorial class representing the shifted shuffle
          product between words ``self`` and ``other``. This is the same as the
          shuffle product of ``self`` with the word obtained from ``other`` by
          incrementing its values (i.e. its letters) by the given ``shift``.
  
          INPUT:
  
          - ``other`` -- finite word over the integers
          - ``shift`` -- integer or ``None`` (default: ``None``) added to each letter of
            ``other``. When ``shift`` is ``None``, it is replaced by ``self.length()``
  
          OUTPUT:
  
          combinatorial class of shifted shuffle products of ``self`` and ``other``
  
          EXAMPLES::
  
              sage: w = Word([0,1,1])
              sage: sp = w.shifted_shuffle(w); sp
              Shuffle product of word: 011 and word: 344
              sage: sp = w.shifted_shuffle(w, 2); sp
              Shuffle product of word: 011 and word: 233
              sage: sp.cardinality()
              20
              sage: WordOptions(identifier='')
              sage: sp.list()
              [011233, 012133, 012313, 012331, 021133, 021313, 021331, 023113, 023131, 023311, 201133, 201313, 201331, 203113, 203131, 203311, 230113, 230131, 230311, 233011]
              sage: WordOptions(identifier='word: ')
              sage: y = Word('aba')
              sage: y.shifted_shuffle(w,2)
              Traceback (most recent call last):
              ...
              ValueError: for shifted shuffle, words must only contain integers as letters
          """
          if any(a not in ZZ for a in self) or any(a not in ZZ for a in other):
              raise ValueError("for shifted shuffle, words must only contain integers as letters")
          if shift is None:
              from sage.combinat.words.shuffle_product import ShuffleProduct_shifted
              return ShuffleProduct_shifted(self, other)
          else:
              return self.shuffle(self._parent([x + shift for x in other], check=False))
  
      ######################################################################
      # XXX TODO: The description is inconsistent w.r.t. "W" and "alphabet".
      ######################################################################
      def delta_inv(self, W=None, s=None):
          r"""
          Lift ``self`` via the delta operator to obtain a word containing the
          letters in alphabet (default is ``[0, 1]``). The letters used in the
          construction start with ``s`` (default is ``alphabet[0]``) and cycle
          through alphabet.
  
          INPUT:
  
          - ``alphabet`` -- an iterable
          - ``s`` -- an object in the iterable
  
          EXAMPLES::
  
              sage: W = Words([1, 2])
              sage: W([2, 2, 1, 1]).delta_inv()
              word: 112212
              sage: W([1, 1, 1, 1]).delta_inv(Words('123'))
              word: 1231
              sage: W([2, 2, 1, 1, 2]).delta_inv(s=2)
              word: 22112122
          """
          alphabet = [1, 2] if W is None else W.alphabet()
          cycle_alphabet = cycle(alphabet)
          if self.is_empty():
              return Words(alphabet)()
          if s is None:
              s = next(cycle_alphabet)
          else:
              if s not in alphabet:
                  raise ValueError("starting letter not in alphabet")
              t = next(cycle_alphabet)
              while t != s:
                  t = next(cycle_alphabet)
          w = []
          for i in self:
              w.extend([s] * i)
              s = next(cycle_alphabet)
          return Words(alphabet)(w)
  
      def delta(self):
          r"""
          Return the image of ``self`` under the delta morphism. This is the
          word composed of the length of consecutive runs of the same letter
          in a given word.
  
          EXAMPLES::
  
              sage: W = Words('0123456789')
              sage: W('22112122').delta()
              word: 22112
              sage: W('555008').delta()
              word: 321
              sage: W().delta()
              word:
              sage: Word('aabbabaa').delta()
              word: 22112
  
          """
          if self.is_empty():
              return Words()([])
          ss = self[0]
          c = 0
          v = list()
          max_c = 0
          for s in self:
              if s == ss:
                  c += 1
                  if c > max_c:
                      max_c = c
              else:
                  v.append(c)
                  ss = s
                  c = 1
          v.append(c)
          return Words(list(range(1, 1 + max_c)))(v)
  
      # TODO. Decide whether delta_derivate* really need W.alphabet().last()....
      # RENAME: Should "derivate" be derivative?!
  
      def delta_derivate(self, W=None):
          r"""
          Return the derivative under delta for ``self``.
  
          EXAMPLES::
  
              sage: W = Words('12')
              sage: W('12211').delta_derivate()
              word: 22
              sage: W('1').delta_derivate(Words([1]))
              word: 1
              sage: W('2112').delta_derivate()
              word: 2
              sage: W('2211').delta_derivate()
              word: 22
              sage: W('112').delta_derivate()
              word: 2
              sage: W('11222').delta_derivate(Words([1, 2, 3]))
              word: 3
          """
          d = self.delta()
          if len(d) == 0:
              return d
          if W is None:
              W = d.parent()
          if d[0] != W.alphabet().last():
              d = d[1:]
          if d[-1] != W.alphabet().last():
              d = d[:-1]
          return d
  
      def delta_derivate_left(self, W=None):
          r"""
          Return the derivative under delta for ``self``.
  
          EXAMPLES::
  
              sage: W = Words('12')
              sage: W('12211').delta_derivate_left()
              word: 22
              sage: W('1').delta_derivate_left(Words([1]))
              word: 1
              sage: W('2112').delta_derivate_left()
              word: 21
              sage: W('2211').delta_derivate_left()
              word: 22
              sage: W('112').delta_derivate_left()
              word: 21
              sage: W('11222').delta_derivate_left(Words([1, 2, 3]))
              word: 3
          """
          d = self.delta()
          if len(d) == 0:
              return d
          if W is None:
              W = d.parent()
          if d[0] != W.alphabet().last():
              d = d[1:]
          return d
  
      def delta_derivate_right(self, W=None):
          r"""
          Return the right derivative under delta for ``self``.
  
          EXAMPLES::
  
              sage: W = Words('12')
              sage: W('12211').delta_derivate_right()
              word: 122
              sage: W('1').delta_derivate_right(Words([1]))
              word: 1
              sage: W('2112').delta_derivate_right()
              word: 12
              sage: W('2211').delta_derivate_right()
              word: 22
              sage: W('112').delta_derivate_right()
              word: 2
              sage: W('11222').delta_derivate_right(Words([1, 2, 3]))
              word: 23
          """
          d = self.delta()
          if len(d) == 0:
              return d
          if W is None:
              W = d.parent()
          if d[-1] != W.alphabet().last():
              d = d[:-1]
          return d
  
      def phi(self):
          r"""
          Apply the phi function to ``self`` and return the result. This is
          the word obtained by taking the first letter of the words obtained
          by iterating delta on ``self``.
  
          OUTPUT:
  
          a word -- the result of the phi function
  
          EXAMPLES::
  
              sage: W = Words([1, 2])
              sage: W([2,2,1,1,2,1,2,2,1,2,2,1,1,2]).phi()
              word: 222222
              sage: W([2,1,2,2,1,2,2,1,2,1]).phi()
              word: 212113
              sage: W().phi()
              word:
              sage: Word([2,1,2,2,1,2,2,1,2,1]).phi()
              word: 212113
              sage: Word([2,3,1,1,2,1,2,3,1,2,2,3,1,2]).phi()
              word: 21215
              sage: Word("aabbabaabaabba").phi()
              word: a22222
              sage: w = Word([2,3,1,1,2,1,2,3,1,2,2,3,1,2])
  
          REFERENCES:
  
          -   S. Brlek, A. Ladouceur, A note on differentiable palindromes,
              Theoret. Comput. Sci. 302 (2003) 167--178.
          -   S. Brlek, S. Dulucq, A. Ladouceur, L. Vuillon, Combinatorial
              properties of smooth infinite words, Theoret. Comput. Sci. 352
              (2006) 306--317.
          """
          if self.is_empty():
              return self
          v = [self[0]]
          m = self.delta()
          while m.length() > 1:
              v.append(m[0])
              m = m.delta()
          v.append(m[0])
          return Words()(v)
  
      def phi_inv(self, W=None):
          r"""
          Apply the inverse of the phi function to ``self``.
  
          INPUT:
  
          - ``self`` -- a word over the integers
          - ``W`` -- a parent object of words defined over integers
  
          OUTPUT:
  
          a word -- the inverse of the phi function
  
          EXAMPLES::
  
              sage: W = Words([1, 2])
              sage: W([2, 2, 2, 2, 1, 2]).phi_inv()
              word: 22112122
              sage: W([2, 2, 2]).phi_inv(Words([2, 3]))
              word: 2233
          """
          if W is None:
              W = self.parent()
          if self.is_empty():
              return W()
          v = self.parent()((self[-1],), check=False)
          for i in range(self.length()-2, -1, -1):
              v = v.delta_inv(W, self[i])
          return v
  
      def _phi_inv_tab(self, tab):
          r"""
          Specialized version of ``phi_inv()`` for long or incremental words.
  
          TESTS::
  
              sage: Words([1, 2])([1, 1, 2, 2])._phi_inv_tab([2])
              word: 12211
          """
          res = self.delta_inv(s=tab[0])
          res = res[1:]
          for i in range(1, len(tab)):
              res = res.delta_inv(s=tab[i])
          return res
  
      def is_smooth_prefix(self):
          r"""
          Return ``True`` if ``self`` is the prefix of a smooth word, and ``False``
          otherwise.
  
          Let `A_k = \{1, \ldots ,k\}`, `k \geq 2`. An infinite word `w` in
          `A_k^\omega` is said to be *smooth* if and only if for all positive
          integers `m`, `\Delta^m(w)` is in `A_k^\omega`, where `\Delta(w)` is
          the word obtained from `w` by composing the length of consecutive
          runs of the same letter in `w`. See for instance [1] and [2].
  
          INPUT:
  
          - ``self`` -- must be a word over the integers to get something other
            than ``False``
  
          OUTPUT:
  
          boolean -- whether ``self`` is a smooth prefix or not
  
          EXAMPLES::
  
              sage: W = Words([1, 2])
              sage: W([1, 1, 2, 2, 1, 2, 1, 1]).is_smooth_prefix()
              True
              sage: W([1, 2, 1, 2, 1, 2]).is_smooth_prefix()
              False
  
          REFERENCES:
  
          -   [1] S. Brlek, A. Ladouceur, A note on differentiable palindromes,
              Theoret. Comput. Sci. 302 (2003) 167--178.
          -   [2] S. Brlek, S. Dulucq, A. Ladouceur, L. Vuillon, Combinatorial
              properties of smooth infinite words, Theoret. Comput. Sci. 352
              (2006) 306--317.
          """
          m = self
          W = self.parent()
          while m.length() > 1:
              m = m.delta_derivate_right()
              if not all(a in W.alphabet() for a in m.letters()):
                  return False
          return True
  
      def letters(self):
          r"""
          Return the list of letters that appear in this word, listed in the
          order of first appearance.
  
          EXAMPLES::
  
              sage: Word([0,1,1,0,1,0,0,1]).letters()
              [0, 1]
              sage: Word("cacao").letters()
              ['c', 'a', 'o']
  
          TESTS::
  
              sage: Word().letters()
              []
          """
          seen = set()
          res = []
          for x in self:
              if x not in seen:
                  res.append(x)
                  seen.add(x)
          return res
  
      def standard_factorization(self):
          r"""
          Return the standard factorization of ``self``.
  
          The *standard factorization* of a word `w` of length greater than
          `1` is the factorization `w = uv` where `v` is the longest proper
          suffix of `w` that is a Lyndon word.
  
          Note that if `w` is a Lyndon word of length greater than `1` with
          standard factorization `w = uv`, then `u` and `v` are also Lyndon
          words and `u < v`.
  
          See for instance [1], [2] and [3].
  
          INPUT:
  
          - ``self`` -- finite word of length greater than `1`
  
          OUTPUT:
  
          `2`-tuple `(u, v)`
  
          EXAMPLES::
  
              sage: Words('01')('0010110011').standard_factorization()
              (word: 001011, word: 0011)
              sage: Words('123')('1223312').standard_factorization()
              (word: 12233, word: 12)
              sage: Word([3,2,1]).standard_factorization()
              (word: 32, word: 1)
  
          ::
  
              sage: w = Word('0010110011',alphabet='01')
              sage: w.standard_factorization()
              (word: 001011, word: 0011)
              sage: w = Word('0010110011',alphabet='10')
              sage: w.standard_factorization()
              (word: 001011001, word: 1)
              sage: w = Word('1223312',alphabet='123')
              sage: w.standard_factorization()
              (word: 12233, word: 12)
  
          TESTS::
  
              sage: w = Word()
              sage: w.standard_factorization()
              Traceback (most recent call last):
              ...
              ValueError: Standard factorization not defined on words of
              length less than 2
              sage: w = Word('a')
              sage: w.standard_factorization()
              Traceback (most recent call last):
              ...
              ValueError: Standard factorization not defined on words of
              length less than 2
  
          REFERENCES:
  
          -   [1] K.-T. Chen, R.H. Fox, R.C. Lyndon, Free differential calculus,
              IV. The quotient groups of the lower central series, Ann. of Math.
              68 (1958) 81--95.
          -   [2] J.-P. Duval, Factorizing words over an ordered alphabet,
              J. Algorithms 4 (1983) 363--381.
          -   [3] M. Lothaire, Algebraic Combinatorics On Words, vol. 90 of
              Encyclopedia of Mathematics and its Applications, Cambridge
              University Press, U.K., 2002.
          """
          selflen = self.length()
          if selflen < 2:
              raise ValueError("Standard factorization not defined on"
                               " words of length less than 2")
          for l in range(1, selflen):
              suff = self[l:]
              if suff.is_lyndon():
                  return self[:l], suff
  
      def apply_permutation_to_positions(self, permutation):
          r"""
          Return the word obtained by permuting the positions of the letters
          in ``self`` according to the permutation ``permutation``.
  
          EXAMPLES::
  
              sage: w = Words('abcd')('abcd')
              sage: w.apply_permutation_to_positions([2,1,4,3])
              word: badc
              sage: u = Words('dabc')('abcd')
              sage: u.apply_permutation_to_positions([2,1,4,3])
              word: badc
              sage: w.apply_permutation_to_positions(Permutation([2,1,4,3]))
              word: badc
              sage: w.apply_permutation_to_positions(PermutationGroupElement([2,1,4,3]))
              word: badc
              sage: Word([1,2,3,4]).apply_permutation_to_positions([3,4,2,1])
              word: 3421
          """
          from sage.combinat.permutation import Permutation
          from sage.groups.perm_gps.permgroup_element import PermutationGroupElement
          if not isinstance(permutation, Permutation):
              if isinstance(permutation, PermutationGroupElement):
                  permutation = Permutation(permutation.domain())
              else:
                  permutation = Permutation(permutation)
          return self.parent()(permutation.action(self), check=False)
  
      def apply_permutation_to_letters(self, permutation):
          r"""
          Return the word obtained by applying the permutation
          ``permutation`` of the alphabet of ``self`` to each letter of
          ``self``.
  
          EXAMPLES::
  
              sage: w = Words('abcd')('abcd')
              sage: p = [2,1,4,3]
              sage: w.apply_permutation_to_letters(p)
              word: badc
              sage: u = Words('dabc')('abcd')
              sage: u.apply_permutation_to_letters(p)
              word: dcba
              sage: w.apply_permutation_to_letters(Permutation(p))
              word: badc
              sage: w.apply_permutation_to_letters(PermutationGroupElement(p))
              word: badc
          """
          from sage.combinat.permutation import Permutation
          from sage.groups.perm_gps.permgroup_element import PermutationGroupElement
          if not isinstance(permutation, Permutation):
              if isinstance(permutation, PermutationGroupElement):
                  permutation = Permutation(permutation.domain())
              else:
                  permutation = Permutation(permutation)
          alphabet = self.parent().alphabet()
          morphism = dict(zip(alphabet, permutation.action(alphabet)))
          return self.apply_morphism(morphism)
  
      def colored_vector(self, x=0, y=0, width='default', height=1, cmap='hsv', thickness=1, label=None):
          r"""
          Return a vector (Graphics object) illustrating ``self``. Each letter
          is represented by a coloured rectangle.
  
          If the parent of ``self`` is a class of words over a finite alphabet,
          then each letter in the alphabet is assigned a unique colour, and
          this colour will be the same every time this method is called. This
          is especially useful when plotting and comparing words defined on
          the same alphabet.
  
          If the alphabet is infinite, then the letters appearing in the word
          are used as the alphabet.
  
          INPUT:
  
          - ``x`` -- (default: ``0``) bottom left x-coordinate of the vector
          - ``y`` -- (default: ``0``) bottom left y-coordinate of the vector
          - ``width`` -- (default: ``'default'``) width of the vector. By default,
            the width is the length of ``self``.
          - ``height`` -- (default: ``1``) height of the vector
          - ``thickness`` -- (default: ``1``) thickness of the contour
          - ``cmap`` -- (default: ``'hsv'``) color map; for available color map names
            type: ``import matplotlib.cm; list(matplotlib.cm.datad)``
          - ``label`` -- string (default: ``None``) a label to add on the colored vector
  
          OUTPUT:
  
          Graphics
  
          EXAMPLES::
  
              sage: Word(range(20)).colored_vector()
              Graphics object consisting of 21 graphics primitives
              sage: Word(range(100)).colored_vector(0,0,10,1)
              Graphics object consisting of 101 graphics primitives
              sage: Words(range(100))(range(10)).colored_vector()
              Graphics object consisting of 11 graphics primitives
              sage: w = Word('abbabaab')
              sage: w.colored_vector()
              Graphics object consisting of 9 graphics primitives
              sage: w.colored_vector(cmap='autumn')
              Graphics object consisting of 9 graphics primitives
              sage: Word(range(20)).colored_vector(label='Rainbow')
              Graphics object consisting of 23 graphics primitives
  
          When two words are defined under the same parent, same letters are
          mapped to same colors::
  
              sage: W = Words(range(20))
              sage: w = W(range(20))
              sage: y = W(range(10,20))
              sage: y.colored_vector(y=1, x=10) + w.colored_vector()
              Graphics object consisting of 32 graphics primitives
  
          TESTS:
  
          The empty word::
  
              sage: Word().colored_vector()
              Graphics object consisting of 1 graphics primitive
              sage: Word().colored_vector(label='empty')
              Graphics object consisting of 3 graphics primitives
  
          Unknown cmap::
  
              sage: Word(range(100)).colored_vector(cmap='jolies')
              Traceback (most recent call last):
              ...
              RuntimeError: Color map jolies not known
              sage: Word(range(100)).colored_vector(cmap='__doc__')
              Traceback (most recent call last):
              ...
              RuntimeError: Color map __doc__ not known
          """
          #Recognize the color map
          import matplotlib.cm as cm
          from matplotlib.colors import LinearSegmentedColormap as C
          key_error = False
          try:
              mpl_cmap = cm.__dict__[cmap]
          except KeyError:
              key_error = True
  
          if key_error or not isinstance(mpl_cmap, C):
              possibilities = ', '.join([str(x) for x in cm.__dict__.keys() if \
                                         isinstance(cm.__dict__[x], C)])
              import sage.misc.misc
              sage.misc.misc.verbose("The possible color maps include: %s"%possibilities, level=0)
              raise RuntimeError("Color map %s not known"%cmap)
  
          #Drawing the colored vector...
          from sage.plot.plot import polygon,line,text
  
          #The default width of the vector
          if width == 'default':
              width = self.length()
  
          #The black frame of the vector
          ymax = y + height
          L = [(x,y), (x+width,y), (x+width,ymax), (x,ymax), (x,y)]
          rep = line(L, rgbcolor=(0,0,0), thickness=thickness)
  
          #The label
          if not label is None:
              hl = height/2.0 # height of the label rectangle
              ymax2 = ymax + hl
              rep += text(str(label), (x+width/2.0, ymax + hl/2.0), rgbcolor=(1,0,0))
              L = [(x,ymax), (x+width,ymax), (x+width,ymax2), (x,ymax2), (x,ymax)]
              rep += line(L, rgbcolor=(0,0,0), thickness=thickness)
  
          #base : the width of each rectangle
          base = width / float(self.length()) if not self.is_empty() else None
  
          #A colored rectangle for each letter
          dim = self.parent().alphabet().cardinality()
          if dim is Infinity:
              ordered_alphabet = sorted(self.letters(),
                                        key=self.parent().sortkey_letters)
              dim = float(len(ordered_alphabet))
          else:
              ordered_alphabet = self.parent().alphabet()
              dim = float(self.parent().alphabet().cardinality())
          letter_to_integer_dict = dict((a,i) for (i,a) in
                  enumerate(ordered_alphabet))
          xp = x
          for a in self:
              i = letter_to_integer_dict[a]
              xq = xp + base
              L = [(xp,y), (xq,y), (xq,ymax), (xp,ymax) ]
              rgbcolor = mpl_cmap( i / dim ) [:3]
              rep += polygon(L, rgbcolor = rgbcolor)
              xp = xq
          rep.axes(False)
          return rep
  
      def is_square(self):
          r"""
          Return ``True`` if ``self`` is a square, and ``False`` otherwise.
  
          EXAMPLES::
  
              sage: Word([1,0,0,1]).is_square()
              False
              sage: Word('1212').is_square()
              True
              sage: Word('1213').is_square()
              False
              sage: Word('12123').is_square()
              False
              sage: Word().is_square()
              True
          """
          if self.length() % 2 != 0:
              return False
          else:
              l = self.length() // 2
              return self[:l] == self[l:]
  
      def is_square_free(self):
          r"""
          Return ``True`` if ``self`` does not contain squares, and ``False``
          otherwise.
  
          EXAMPLES::
  
              sage: Word('12312').is_square_free()
              True
              sage: Word('31212').is_square_free()
              False
              sage: Word().is_square_free()
              True
  
          TESTS:
  
          We make sure that :trac:`8490` is fixed::
  
              sage: Word('11').is_square_free()
              False
              sage: Word('211').is_square_free()
              False
              sage: Word('3211').is_square_free()
              False
          """
          L = self.length()
          if L < 2:
              return True
          for start in range(0, L-1):
              for end in range(start+2, L+1, 2):
                  if self[start:end].is_square():
                      return False
          return True
  
      def is_cube(self):
          r"""
          Return ``True`` if ``self`` is a cube, and ``False`` otherwise.
  
          EXAMPLES::
  
              sage: Word('012012012').is_cube()
              True
              sage: Word('01010101').is_cube()
              False
              sage: Word().is_cube()
              True
              sage: Word('012012').is_cube()
              False
          """
          if self.length() % 3 != 0:
              return False
          l = self.length() // 3
          return self[:l] == self[l:2*l] == self[2*l:]
  
      def is_cube_free(self):
          r"""
          Return ``True`` if ``self`` does not contain cubes, and ``False`` otherwise.
  
          EXAMPLES::
  
              sage: Word('12312').is_cube_free()
              True
              sage: Word('32221').is_cube_free()
              False
              sage: Word().is_cube_free()
              True
  
          TESTS:
  
          We make sure that :trac:`8490` is fixed::
  
              sage: Word('111').is_cube_free()
              False
              sage: Word('2111').is_cube_free()
              False
              sage: Word('32111').is_cube_free()
              False
          """
          L = self.length()
          if L < 3:
              return True
          for start in range(0, L - 2):
              for end in range(start+3, L+1, 3):
                  if self[start:end].is_cube():
                      return False
          return True
  
      def to_monoid_element(self):
          """
          Return ``self`` as an element of the free monoid with the same alphabet
          as ``self``.
  
          EXAMPLES::
  
              sage: w = Word('aabb')
              sage: w.to_monoid_element()
              a^2*b^2
              sage: W = Words('abc')
              sage: w = W(w)
              sage: w.to_monoid_element()
              a^2*b^2
  
          TESTS:
  
          Check that ``w == w.to_monoid_element().to_word()``::
  
              sage: all(w.to_monoid_element().to_word() == w for i in range(6) for w in Words('abc', i))
              True
          """
          from sage.monoids.free_monoid import FreeMonoid
          try:
              l = list(self.parent().alphabet())
          except AttributeError:
              l = self.letters()
          M = FreeMonoid(len(l), l)
          return M(self)
  
      def is_christoffel(self):
          r"""
          Return ``True`` if ``self`` is a Christoffel word, and ``False`` otherwise.
  
          The *Christoffel word* of slope `p/q` is obtained from the Cayley
          graph of `\ZZ/(p+q)\ZZ` with generator `q` as follows. If `u
          \rightarrow v` is an edge in the Cayley graph, then, `v = u + p
          \mod{p+q}`. Let `a`,`b` be the alphabet of `w`. Label the edge
          `u \rightarrow v` by `a` if `u < v` and `b` otherwise. The Christoffel
          word is the word obtained by reading the edge labels along the cycle
          beginning from `0`.
  
          Equivalently, `w` is a Christoffel word iff `w` is a symmetric
          non-empty word and `w[1:n-1]` is a palindrome.
  
          See for instance [1]_ and [2]_.
  
          INPUT:
  
          - ``self`` -- word
  
          OUTPUT:
  
          boolean -- ``True`` if ``self`` is a Christoffel word,
          ``False`` otherwise.
  
          EXAMPLES::
  
              sage: Word('00100101').is_christoffel()
              True
              sage: Word('aab').is_christoffel()
              True
              sage: Word().is_christoffel()
              False
              sage: Word('123123123').is_christoffel()
              False
              sage: Word('00100').is_christoffel()
              False
              sage: Word('0').is_christoffel()
              True
  
          TESTS::
  
              sage: words.LowerChristoffelWord(5,4).is_christoffel()
              True
              sage: words.UpperChristoffelWord(5,4).is_christoffel()
              True
              sage: Word('aaaaaaaaa').is_christoffel()
              False
  
          REFERENCES:
  
          .. [1]  Jean Berstel. Sturmian and episturmian words (a survey of
              some recent results). In S. Bozapalidis and G. Rahonis, editors,
              CAI 2007,volume 4728 of Lecture Notes in Computer Science,
              pages 23-47. Springer-Verlag, 2007.
          .. [2] \J. Berstel, A. Lauve, C. R., F. Saliola, Combinatorics on
              words: Christoffel words and repetitions in words, CRM Monograph
              Series, 27. American Mathematical Society, Providence, RI, 2009.
              xii+147 pp. ISBN: 978-0-8218-4480-9
  
          """
          if len(self) == 0 or len(self.letters()) > 2 or (self.is_palindrome() and len(self) > 1):
              return False
          elif self.is_symmetric() and self[1:len(self) - 1].is_palindrome():
              return True
          else:
              return False
  
  
  #######################################################################
  
  class CallableFromListOfWords(tuple):
      r"""
      A class to create a callable from a list of words. The concatenation of
      a list of words is obtained by creating a word from this callable.
      """
      def __new__(cls, words):
          r"""
          TESTS::
  
              sage: from sage.combinat.words.finite_word import CallableFromListOfWords
              sage: w,u,x = Word([1,2,3]),Word([4,5]),Word([6,7,8])
              sage: f = CallableFromListOfWords([w,u,x]); f
              (word: 123, word: 45, word: 678)
              sage: f == loads(dumps(f))
              True
          """
          l = []
          for w in words:
              from .word_infinite_datatypes import WordDatatype_callable
              if isinstance(w, WordDatatype_callable) and \
                      isinstance(w._func, CallableFromListOfWords):
                  l.extend(w._func)
              else:
                  l.append(w)
          return tuple.__new__(cls, l)
  
      def __call__(self, i):
          r"""
          Return the character at position ``i``.
  
          TESTS::
  
              sage: from sage.combinat.words.finite_word import CallableFromListOfWords
              sage: w,u,x = Word([1,2,3]),Word([4,5]),Word([6,7,8])
              sage: f = CallableFromListOfWords([w,u,x])
              sage: [f(i) for i in range(8)]
              [1, 2, 3, 4, 5, 6, 7, 8]
          """
          j = i
          for c in self:
              if (j - c.length() < 0):
                  return c[j]
              j -= c.length()
          raise IndexError("index (=%s) out of range" % i)
  
  class Factorization(list):
      r"""
      A list subclass having a nicer representation for factorization of words.
  
      TESTS::
  
          sage: f = sage.combinat.words.finite_word.Factorization()
          sage: f == loads(dumps(f))
          True
      """
      def __repr__(self):
          r"""
          Return a string representation of the object.
  
          TESTS::
  
              sage: sage.combinat.words.finite_word.Factorization()
              ()
              sage: sage.combinat.words.finite_word.Factorization([Word('ab'), Word('ba')])
              (ab, ba)
          """
          return '(%s)' % ', '.join(w.string_rep() for w in self)
  
  #######################################################################
  
  def evaluation_dict(w):
      r"""
      Return a dictionary keyed by the letters occurring in ``w`` with
      values the number of occurrences of the letter.
  
      INPUT:
  
      - ``w`` -- a word
  
      TESTS::
  
          sage: from sage.combinat.words.finite_word import evaluation_dict
          sage: evaluation_dict([2,1,4,2,3,4,2])
          {1: 1, 2: 3, 3: 1, 4: 2}
          sage: evaluation_dict('badbcdb')
          {'a': 1, 'b': 3, 'c': 1, 'd': 2}
          sage: evaluation_dict([])
          {}
  
      ::
  
          sage: evaluation_dict('1213121') # keys appear in random order
          {'1': 4, '2': 2, '3': 1}
  
      """
      d = defaultdict(int)
      for a in w:
          d[a] += 1
      return dict(d)
  
  