text: # -*- coding: utf-8 -*-
  r"""
  Dyck Words
  
  A class of an object enumerated by the
  :func:`Catalan numbers<sage.combinat.combinat.catalan_number>`,
  see [Sta-EC2]_, [StaCat98]_ for details.
  
  AUTHORS:
  
  - Mike Hansen
  
  - Dan Drake (2008--05-30): DyckWordBacktracker support
  
  - Florent Hivert (2009--02-01): Bijections with NonDecreasingParkingFunctions
  
  - Christian Stump (2011--12): added combinatorial maps and statistics
  
  - Mike Zabrocki:
  
    * (2012--10): added pretty print, characteristic function, more functions
    * (2013--01): added inverse of area/dinv, bounce/area map
  
  - Jean--Baptiste Priez, Travis Scrimshaw (2013--05-17): Added ASCII art
  
  - Travis Scrimshaw (2013--07-09): Removed ``CombinatorialClass`` and added
    global options.
  
  REFERENCES:
  
  .. [Sta-EC2] Richard P. Stanley.
     *Enumerative Combinatorics*, Volume 2.
     Cambridge University Press, 2001.
  
  .. [StaCat98] Richard Stanley. *Exercises on Catalan and Related Numbers
     excerpted from Enumerative Combinatorics, vol. 2 (CUP 1999)*,
     version of 23 June 1998.
     http://www-math.mit.edu/~rstan/ec/catalan.pdf
  
  .. [Hag2008] James Haglund. *The* `q,t` -- *Catalan Numbers and the
     Space of Diagonal Harmonics:
     With an Appendix on the Combinatorics of Macdonald Polynomials*.
     University of Pennsylvania, Philadelphia -- AMS, 2008, 167 pp.
  """
  
  #*****************************************************************************
  #       Copyright (C) 2007 Mike Hansen <mhansen@gmail.com>,
  #
  #  Distributed under the terms of the GNU General Public License (GPL)
  #
  #    This code is distributed in the hope that it will be useful,
  #    but WITHOUT ANY WARRANTY; without even the implied warranty of
  #    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
  #    General Public License for more details.
  #
  #  The full text of the GPL is available at:
  #
  #                  http://www.gnu.org/licenses/
  #*****************************************************************************
  
  from combinat import CombinatorialElement, catalan_number
  from sage.combinat.combinatorial_map import combinatorial_map
  from backtrack import GenericBacktracker
  
  from sage.structure.global_options import GlobalOptions
  from sage.structure.parent import Parent
  from sage.structure.unique_representation import UniqueRepresentation
  from sage.categories.finite_enumerated_sets import FiniteEnumeratedSets
  from sage.categories.infinite_enumerated_sets import InfiniteEnumeratedSets
  from sage.categories.all import Posets
  
  from sage.rings.all import ZZ, QQ
  from sage.combinat.permutation import Permutation, Permutations
  from sage.combinat.words.word import Word
  from sage.combinat.alternating_sign_matrix import AlternatingSignMatrices
  from sage.misc.latex import latex
  
  
  DyckWordOptions = GlobalOptions(name='Dyck words',
      doc=r"""
      Set and display the global options for Dyck words. If no parameters
      are set, then the function returns a copy of the options dictionary.
  
      The ``options`` to Dyck words can be accessed as the method
      :obj:`DyckWords.global_options` of :class:`DyckWords` and
      related parent classes.
      """,
      end_doc=r"""
      EXAMPLES::
  
          sage: D = DyckWord([1, 1, 0, 1, 0, 0])
          sage: D
          [1, 1, 0, 1, 0, 0]
          sage: DyckWords.global_options(display="lattice")
          sage: D
             ___
           _| x
          | x  .
          |  . .
          sage: DyckWords.global_options(diagram_style="line")
          sage: D
           /\/\
          /    \
          sage: DyckWords.global_options.reset()
      """,
      display=dict(default="list",
                   description='Specifies how Dyck words should be printed',
                   values=dict(list='displayed as a list',
                               lattice='displayed on the lattice defined by ``diagram_style``'),
                   case_sensitive=False),
      ascii_art=dict(default="path",
                     description='Specifies how the ascii art of Dyck words should be printed',
                     values=dict(path="Using the path string",
                                 pretty_output="Using pretty printing"),
                     alias=dict(pretty_print="pretty_output", path_string="path"),
                     case_sensitive=False),
      diagram_style=dict(default="grid",
                         values=dict(grid='printing as paths on a grid using N and E steps',
                                     line='printing as paths on a line using NE and SE steps',),
                         alias={'N-E': 'grid', 'NE-SE': 'line'},
                         case_sensitive=False),
      latex_tikz_scale=dict(default=1,
                            description='The default value for the tikz scale when latexed',
                            checker=lambda x: True),  # More trouble than it's worth to check
      latex_diagonal=dict(default=False,
                          description='The default value for displaying the diagonal when latexed',
                          checker=lambda x: isinstance(x, bool)),
      latex_line_width_scalar=dict(default=2,
                                   description='The default value for the line width as a'
                                               'multiple of the tikz scale when latexed',
                                   checker=lambda x: True),  # More trouble than it's worth to check
      latex_color=dict(default="black",
                       description='The default value for the color when latexed',
                       checker=lambda x: isinstance(x, str)),
      latex_bounce_path=dict(default=False,
                             description='The default value for displaying the bounce path when latexed',
                             checker=lambda x: isinstance(x, bool)),
      latex_peaks=dict(default=False,
                       description='The default value for displaying the peaks when latexed',
                       checker=lambda x: isinstance(x, bool)),
      latex_valleys=dict(default=False,
                        description='The default value for displaying the valleys when latexed',
                        checker=lambda x: isinstance(x, bool)),
  )
  
  open_symbol = 1
  close_symbol = 0
  
  
  def replace_parens(x):
      r"""
      A map sending ``'('`` to ``open_symbol`` and ``')'`` to
      ``close_symbol``, and raising an error on any input other than
      ``'('``  and ``')'``. The values of the constants ``open_symbol``
      and ``close_symbol`` are subject to change.
  
      This is the inverse map of :func:`replace_symbols`.
  
      INPUT:
  
      - ``x`` -- either an opening or closing parenthesis
  
      OUTPUT:
  
      - If ``x`` is an opening parenthesis, replace ``x`` with the
        constant ``open_symbol``.
  
      - If ``x`` is a closing parenthesis, replace ``x`` with the
        constant ``close_symbol``.
  
      - Raise a ``ValueError`` if ``x`` is neither an opening nor a
        closing parenthesis.
  
      .. SEEALSO:: :func:`replace_symbols`
  
      EXAMPLES::
  
          sage: from sage.combinat.dyck_word import replace_parens
          sage: replace_parens('(')
          1
          sage: replace_parens(')')
          0
          sage: replace_parens(1)
          Traceback (most recent call last):
          ...
          ValueError
      """
      if x == '(':
          return open_symbol
      elif x == ')':
          return close_symbol
      else:
          raise ValueError
  
  
  def replace_symbols(x):
      r"""
      A map sending ``open_symbol`` to ``'('`` and ``close_symbol`` to ``')'``,
      and raising an error on any input other than ``open_symbol`` and
      ``close_symbol``. The values of the constants ``open_symbol``
      and ``close_symbol`` are subject to change.
  
      This is the inverse map of :func:`replace_parens`.
  
      INPUT:
  
      - ``x`` -- either ``open_symbol`` or ``close_symbol``.
  
      OUTPUT:
  
      - If ``x`` is ``open_symbol``, replace ``x`` with ``'('``.
  
      - If ``x`` is ``close_symbol``, replace ``x`` with ``')'``.
  
      - If ``x`` is neither ``open_symbol`` nor ``close_symbol``, a
        ``ValueError`` is raised.
  
      .. SEEALSO:: :func:`replace_parens`
  
      EXAMPLES::
  
          sage: from sage.combinat.dyck_word import replace_symbols
          sage: replace_symbols(1)
          '('
          sage: replace_symbols(0)
          ')'
          sage: replace_symbols(3)
          Traceback (most recent call last):
          ...
          ValueError
      """
      if x == open_symbol:
          return '('
      elif x == close_symbol:
          return ')'
      else:
          raise ValueError
  
  
  class DyckWord(CombinatorialElement):
      r"""
      A Dyck word.
  
      A Dyck word is a sequence of open and close symbols such that every close
      symbol has a corresponding open symbol preceding it. That is to say, a
      Dyck word of length `n` is a list with `k` entries 1 and `n - k`
      entries 0 such that the first `i` entries always have at least as many 1s
      among them as 0s. (Here, the 1 serves as the open symbol and the 0 as the
      close symbol.)  Alternatively, the alphabet 1 and 0 can be replaced by
      other characters such as '(' and ')'.
  
      A Dyck word is *complete* if every open symbol moreover has a corresponding
      close symbol.
  
      A Dyck word may also be specified by either a noncrossing partition or
      by an area sequence or the sequence of heights.
  
      A Dyck word may also be thought of as a lattice path in the `\mathbb{Z}^2`
      grid, starting at the origin `(0,0)`, and with steps in the North
      `N = (0,1)` and east `E = (1,0)` directions such that it does not pass
      below the `x = y` diagonal. The diagonal is referred to as the "main
      diagonal" in the documentation. A North step is represented by a 1 in
      the list and an East step is represented by a 0.
  
      Equivalently, the path may be represented with steps in
      the `NE = (1,1)` and the `SE = (1,-1)` direction such that it does not
      pass below the horizontal axis.
  
      .. PLOT::
          :width: 400 px
  
          d = DyckWord([1,0,1,1,1,1,0,1,0,0,1,0,1,1,0,1,0,1,1,0,0,0,0,0])
          sphinx_plot(d.plot(aspect_ratio=1))
  
      A path representing a Dyck word (either using `N` and `E` steps, or
      using `NE` and `SE` steps) is called a Dyck path.
  
      EXAMPLES::
  
          sage: dw = DyckWord([1, 0, 1, 0]); dw
          [1, 0, 1, 0]
          sage: print dw
          ()()
          sage: print dw.height()
          1
          sage: dw.to_noncrossing_partition()
          [[1], [2]]
  
      ::
  
          sage: DyckWord('()()')
          [1, 0, 1, 0]
          sage: DyckWord('(())')
          [1, 1, 0, 0]
          sage: DyckWord('((')
          [1, 1]
          sage: DyckWord('')
          []
  
      ::
  
          sage: DyckWord(noncrossing_partition=[[1],[2]])
          [1, 0, 1, 0]
          sage: DyckWord(noncrossing_partition=[[1,2]])
          [1, 1, 0, 0]
          sage: DyckWord(noncrossing_partition=[])
          []
  
      ::
  
          sage: DyckWord(area_sequence=[0,0])
          [1, 0, 1, 0]
          sage: DyckWord(area_sequence=[0,1])
          [1, 1, 0, 0]
          sage: DyckWord(area_sequence=[0,1,2,2,0,1,1,2])
          [1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0]
          sage: DyckWord(area_sequence=[])
          []
  
      ::
  
          sage: DyckWord(heights_sequence=(0,1,0,1,0))
          [1, 0, 1, 0]
          sage: DyckWord(heights_sequence=(0,1,2,1,0))
          [1, 1, 0, 0]
          sage: DyckWord(heights_sequence=(0,))
          []
  
      ::
  
          sage: print DyckWord([1,0,1,1,0,0]).to_path_string()
             /\
          /\/  \
          sage: DyckWord([1,0,1,1,0,0]).pretty_print()
             ___
            | x
           _|  .
          |  . .
      """
      @staticmethod
      def __classcall_private__(cls, dw=None, noncrossing_partition=None,
                                area_sequence=None, heights_sequence=None,
                                catalan_code=None):
          """
          Return an element with the appropriate parent.
  
          EXAMPLES::
  
              sage: DyckWord([1,0,1,1,0,0])
              [1, 0, 1, 1, 0, 0]
              sage: DyckWord(heights_sequence=(0,1,2,1,0))
              [1, 1, 0, 0]
              sage: DyckWord(noncrossing_partition=[[1],[2]])
              [1, 0, 1, 0]
          """
          if dw is None:
              if catalan_code is not None:
                  return CompleteDyckWords_all().from_Catalan_code(catalan_code)
              if area_sequence is not None:
                  return CompleteDyckWords_all().from_area_sequence(area_sequence)
              if noncrossing_partition is not None:
                  return CompleteDyckWords_all().from_noncrossing_partition(noncrossing_partition)
              if heights_sequence is not None:
                  if heights_sequence[-1] == 0:
                      P = CompleteDyckWords_all()
                  else:
                      P = DyckWords_all()
                  return P.from_heights(heights_sequence)
  
              raise ValueError("You have not specified a Dyck word.")
  
          if isinstance(dw, str):
              l = [replace_parens(_) for _ in dw]
          else:
              l = dw
  
          if isinstance(l, DyckWord):
              return l
  
          # CS: what happens here? there is a loop after a return (which is thus never used)
          #elif l in DyckWords() or is_a(l):
              #return DyckWord(l)
              #for opt in l._latex_options:
                  #if opt not in latex_options:
                      #latex_options[opt] = l._latex_options[opt]
              #return DyckWord(l,latex_options=latex_options)
          if l in CompleteDyckWords_all():
              return CompleteDyckWords_all()(l)
          if is_a(l):
              return DyckWords_all()(l)
  
          raise ValueError("invalid Dyck word")
  
      def __init__(self, parent, l, latex_options={}):
          r"""
          TESTS::
  
              sage: DW = DyckWords(complete=False).from_heights((0,))
              sage: TestSuite(DW).run()
              sage: DW = DyckWords(complete=False).min_from_heights((0,))
              sage: TestSuite(DW).run()
              sage: DW = DyckWords().from_Catalan_code([])
              sage: TestSuite(DW).run()
              sage: DW = DyckWords().from_area_sequence([])
              sage: TestSuite(DW).run()
          """
          CombinatorialElement.__init__(self, parent, l)
          self._latex_options = dict(latex_options)
  
      _has_2D_print = False
  
      def set_latex_options(self, D):
          r"""
          Set the latex options for use in the ``_latex_`` function.  The
          default values are set in the ``__init__`` function.
  
          - ``tikz_scale`` -- (default: 1) scale for use with the tikz package.
  
          - ``diagonal`` -- (default: ``False``) boolean value to draw the
            diagonal or not.
  
          - ``line width`` -- (default: 2*``tikz_scale``) value representing the
            line width.
  
          - ``color`` -- (default: black) the line color.
  
          - ``bounce path`` -- (default: ``False``) boolean value to indicate
            if the bounce path should be drawn.
  
          - ``peaks`` -- (default: ``False``) boolean value to indicate if the
            peaks should be displayed.
  
          - ``valleys`` -- (default: ``False``) boolean value to indicate if the
            valleys should be displayed.
  
          INPUT:
  
          - ``D`` -- a dictionary with a list of latex parameters to change
  
          EXAMPLES::
  
              sage: D = DyckWord([1,0,1,0,1,0])
              sage: D.set_latex_options({"tikz_scale":2})
              sage: D.set_latex_options({"valleys":True, "color":"blue"})
          """
          for opt in D:
              self._latex_options[opt] = D[opt]
  
      def latex_options(self):
          r"""
          Return the latex options for use in the ``_latex_`` function as a
          dictionary. The default values are set using the global options.
  
          - ``tikz_scale`` -- (default: 1) scale for use with the tikz package.
  
          - ``diagonal`` -- (default: ``False``) boolean value to draw the
            diagonal or not.
  
          - ``line width`` -- (default: 2*``tikz_scale``) value representing the
            line width.
  
          - ``color`` -- (default: black) the line color.
  
          - ``bounce path`` -- (default: ``False``) boolean value to indicate
            if the bounce path should be drawn.
  
          - ``peaks`` -- (default: ``False``) boolean value to indicate if the
            peaks should be displayed.
  
          - ``valleys`` -- (default: ``False``) boolean value to indicate if the
            valleys should be displayed.
  
          EXAMPLES::
  
              sage: D = DyckWord([1,0,1,0,1,0])
              sage: D.latex_options()
              {'bounce path': False,
               'color': 'black',
               'diagonal': False,
               'line width': 2,
               'peaks': False,
               'tikz_scale': 1,
               'valleys': False}
          """
          d = self._latex_options.copy()
          if "tikz_scale" not in d:
              d["tikz_scale"] = self.parent().global_options["latex_tikz_scale"]
          if "diagonal" not in d:
              d["diagonal"] = self.parent().global_options["latex_diagonal"]
          if "line width" not in d:
              d["line width"] = self.parent().global_options["latex_line_width_scalar"]*d["tikz_scale"]
          if "color" not in d:
              d["color"] = self.parent().global_options["latex_color"]
          if "bounce path" not in d:
              d["bounce path"] = self.parent().global_options["latex_bounce_path"]
          if "peaks" not in d:
              d["peaks"] = self.parent().global_options["latex_peaks"]
          if "valleys" not in d:
              d["valleys"] = self.parent().global_options["latex_valleys"]
          return d
  
      def _repr_(self):
          r"""
          TESTS::
  
              sage: DyckWord([1, 0, 1, 0])
              [1, 0, 1, 0]
              sage: DyckWord([1, 1, 0, 0])
              [1, 1, 0, 0]
              sage: type(DyckWord([]))._has_2D_print = True
              sage: DyckWord([1, 0, 1, 0])
              /\/\
              sage: DyckWord([1, 1, 0, 0])
               /\
              /  \
              sage: type(DyckWord([]))._has_2D_print = False
          """
          if self._has_2D_print:
              return self.to_path_string()
          else:
              return super(DyckWord, self)._repr_()
  
      def _repr_lattice(self, type=None, labelling=None, underpath=True):
          r"""
          See :meth:`pretty_print()`.
  
          TESTS::
  
              sage: print DyckWord(area_sequence=[0,1,0])._repr_lattice(type="NE-SE")
               /\
              /  \/\
              sage: print DyckWord(area_sequence=[0,1,0])._repr_lattice(labelling=[1,3,2],underpath=False)
                   _
               ___|  2
              | x  . 3
              |  . . 1
          """
          if type is None:
              type = self.parent().global_options['diagram_style']
              if type == "grid":
                  type = "N-E"
              elif type == "line":
                  type = "NE-SE"
  
          if type == "NE-SE":
              if labelling is not None or underpath is not True:
                  raise ValueError("The labelling cannot be shown with Northeast-Southeast paths.")
              return self.to_path_string()
          elif type == "N-E":
              alst = self.to_area_sequence()
              n = len(alst)
              if n == 0:
                  return ".\n"
              if labelling is None:
                  labels = [" "]*n
              else:
                  if len(labelling) != n:
                      raise ValueError("The given labelling has the wrong length.")
                  labels = [str(label) for label in labelling]
                  if not underpath:
                      max_length = max(len(label) for label in labels)
                      labels = [lbl.rjust(max_length + 1) for lbl in labels]
  
              length_of_final_fall = list(reversed(self)).index(open_symbol)
              if length_of_final_fall == 0:
                  final_fall = " "
              else:
                  final_fall = " _" + "__"*(length_of_final_fall-1)
              row = "  "*(n - alst[-1]-1) + final_fall + "\n"
              for i in range(n - 1):
                  c = 0
                  row = row + "  "*(n-i-2-alst[-i-2])
                  c += n-i-2-alst[-i-2]
                  if alst[-i-2]+1 != alst[-i-1]:
                      row += " _"
                  c += alst[-i-2] - alst[-i-1]
                  if underpath:
                      row += "__"*(alst[-i-2]-alst[-i-1])+"|" + labels[-1] + "x "*(n-c-2-i) + " ."*i + "\n"
                  else:
                      row += "__"*(alst[-i-2]-alst[-i-1])+"| " + "x "*(n-c-2-i) + " ."*i + labels[-1] + "\n"
                  labels.pop()
              if underpath:
                  row += "|" + labels[-1] + " ."*(n-1) + "\n"
              else:
                  row += "| "+" ."*(n-1) + labels[-1] + "\n"
              return row
          else:
              raise ValueError("The given type (=\s) is not valid." % type)
  
      def _ascii_art_(self):
          r"""
          Return an ASCII art representation of ``self``.
  
          TESTS::
  
              sage: ascii_art(list(DyckWords(3)))
              [                                   /\   ]
              [            /\    /\      /\/\    /  \  ]
              [ /\/\/\, /\/  \, /  \/\, /    \, /    \ ]
          """
          from sage.typeset.ascii_art import AsciiArt
          rep = self.parent().global_options['ascii_art']
          if rep == "path":
              ret = self.to_path_string()
          elif rep == "pretty_output":
              ret = self._repr_lattice()
          return AsciiArt(ret.splitlines(), baseline=0)
  
      def _unicode_art_(self):
          r"""
          Return an unicode art representation of this Dyck word.
  
          EXAMPLES::
  
              sage: unicode_art(list(DyckWords(3)))
              ⎡                                   ╱╲   ⎤
              ⎢            ╱╲    ╱╲      ╱╲╱╲    ╱  ╲  ⎥
              ⎣ ╱╲╱╲╱╲, ╱╲╱  ╲, ╱  ╲╱╲, ╱    ╲, ╱    ╲ ⎦
          """
          from sage.typeset.unicode_art import UnicodeArt
          return UnicodeArt(self.to_path_string(unicode=True).splitlines())
  
      def __str__(self):
          r"""
          Return a string consisting of matched parentheses corresponding to
          the Dyck word.
  
          EXAMPLES::
  
              sage: print DyckWord([1, 0, 1, 0])
              ()()
              sage: print DyckWord([1, 1, 0, 0])
              (())
          """
          if self._has_2D_print:
              return self.to_path_string()
          else:
              return "".join(map(replace_symbols, [x for x in self]))
  
      def to_path_string(self, unicode=False):
          r"""
          A path representation of the Dyck word consisting of steps
          ``/`` and ``\`` .
  
          EXAMPLES::
  
              sage: print DyckWord([1, 0, 1, 0]).to_path_string()
              /\/\
              sage: print DyckWord([1, 1, 0, 0]).to_path_string()
               /\
              /  \
              sage: print DyckWord([1,1,0,1,1,0,0,1,0,1,0,0]).to_path_string()
                  /\
               /\/  \/\/\
              /          \
          """
          if unicode:
              import unicodedata
              space = u' '
              up    = unicodedata.lookup('BOX DRAWINGS LIGHT DIAGONAL UPPER RIGHT TO LOWER LEFT')
              down  = unicodedata.lookup('BOX DRAWINGS LIGHT DIAGONAL UPPER LEFT TO LOWER RIGHT')
          else:
              space = ' '
              up    = '/'
              down  = '\\'
  
          res = [([space]*len(self)) for _ in range(self.height())]
          h = 1
          for i, p in enumerate(self):
              if p == open_symbol:
                  res[-h][i] = up
                  h += 1
              else:
                  h -= 1
                  res[-h][i] = down
          return "\n".join("".join(l) for l in res)
  
      def pretty_print(self, type=None, labelling=None, underpath=True):
          r"""
          Display a DyckWord as a lattice path in the `\ZZ^2` grid.
  
          If the ``type`` is "N-E", then the a cell below the diagonal is
          indicated by a period, whereas a cell below the path but above
          the diagonal is indicated by an x. If a list of labels is
          included, they are displayed along the vertical edges of the
          Dyck path.
  
          If the ``type`` is "NE-SE", then the path is simply printed
          as up steps and down steps.
  
          INPUT:
  
          - ``type`` -- (default: ``None``) can either be:
  
            - ``None`` to use the global option default
            - "N-E" to show ``self`` as a path of north and east steps, or
            - "NE-SE" to show ``self`` as a path of north-east and
              south-east steps.
  
          - ``labelling`` -- (if type is "N-E") a list of labels assigned to
            the up steps in ``self``.
  
          - ``underpath`` -- (if type is "N-E", default:``True``) If ``True``,
            the labelling is shown under the path; otherwise, it is shown to
            the right of the path.
  
          EXAMPLES::
  
              sage: for D in DyckWords(3): D.pretty_print()
                   _
                 _|
               _|  .
              |  . .
                 ___
                | x
               _|  .
              |  . .
                   _
               ___|
              | x  .
              |  . .
                 ___
               _| x
              | x  .
              |  . .
               _____
              | x x
              | x  .
              |  . .
  
          ::
  
              sage: for D in DyckWords(3): D.pretty_print(type="NE-SE")
              /\/\/\
                 /\
              /\/  \
               /\
              /  \/\
               /\/\
              /    \
                /\
               /  \
              /    \
  
          ::
  
              sage: D = DyckWord([1,1,1,0,1,0,0,1,1])
              sage: D.pretty_print()
                    | x x
                 ___| x  .
               _| x x  . .
              | x x  . . .
              | x  . . . .
              |  . . . . .
  
              sage: D = DyckWord([1,1,1,0,1,0,0,1,1,0])
              sage: D.pretty_print()
                     _
                    | x x
                 ___| x  .
               _| x x  . .
              | x x  . . .
              | x  . . . .
              |  . . . . .
  
              sage: D = DyckWord([1,1,1,0,1,0,0,1,1,0,0])
              sage: D.pretty_print()
                     ___
                    | x x
                 ___| x  .
               _| x x  . .
              | x x  . . .
              | x  . . . .
              |  . . . . .
  
          ::
  
              sage: DyckWord(area_sequence=[0,1,0]).pretty_print(labelling=[1,3,2])
                   _
               ___|2
              |3x  .
              |1 . .
  
              sage: DyckWord(area_sequence=[0,1,0]).pretty_print(labelling=[1,3,2],underpath=False)
                   _
               ___|  2
              | x  . 3
              |  . . 1
  
          ::
  
              sage: DyckWord(area_sequence=[0,1,1,2,3,2,3,3,2,0,1,1,2,3,4,2,3]).pretty_print()
                                         _______
                                        | x x x
                                   _____| x x  .
                                  | x x x x  . .
                                  | x x x  . . .
                                  | x x  . . . .
                                 _| x  . . . . .
                                | x  . . . . . .
                           _____|  . . . . . . .
                       ___| x x  . . . . . . . .
                     _| x x x  . . . . . . . . .
                    | x x x  . . . . . . . . . .
                 ___| x x  . . . . . . . . . . .
                | x x x  . . . . . . . . . . . .
                | x x  . . . . . . . . . . . . .
               _| x  . . . . . . . . . . . . . .
              | x  . . . . . . . . . . . . . . .
              |  . . . . . . . . . . . . . . . .
  
              sage: DyckWord(area_sequence=[0,1,1,2,3,2,3,3,2,0,1,1,2,3,4,2,3]).pretty_print(labelling=range(17),underpath=False)
                                         _______
                                        | x x x  16
                                   _____| x x  . 15
                                  | x x x x  . . 14
                                  | x x x  . . . 13
                                  | x x  . . . . 12
                                 _| x  . . . . . 11
                                | x  . . . . . . 10
                           _____|  . . . . . . .  9
                       ___| x x  . . . . . . . .  8
                     _| x x x  . . . . . . . . .  7
                    | x x x  . . . . . . . . . .  6
                 ___| x x  . . . . . . . . . . .  5
                | x x x  . . . . . . . . . . . .  4
                | x x  . . . . . . . . . . . . .  3
               _| x  . . . . . . . . . . . . . .  2
              | x  . . . . . . . . . . . . . . .  1
              |  . . . . . . . . . . . . . . . .  0
  
          ::
  
              sage: DyckWord([]).pretty_print()
              .
          """
          print self._repr_lattice(type, labelling, underpath)
  
      pp = pretty_print
  
      def _latex_(self):
          r"""
          A latex representation of ``self`` using the tikzpicture package.
  
          EXAMPLES::
  
              sage: D = DyckWord([1,0,1,1,1,0,1,1,0,0,0,1,0,0])
              sage: D.set_latex_options({"valleys":True, "peaks":True, "bounce path":True})
              sage: latex(D)
              \vcenter{\hbox{$\begin{tikzpicture}[scale=1]
                \draw[line width=2,color=red,fill=red] (2, 0) circle (0.21);
                \draw[line width=2,color=red,fill=red] (6, 2) circle (0.21);
                \draw[line width=2,color=red,fill=red] (11, 1) circle (0.21);
                \draw[line width=2,color=red,fill=red] (1, 1) circle (0.21);
                \draw[line width=2,color=red,fill=red] (5, 3) circle (0.21);
                \draw[line width=2,color=red,fill=red] (8, 4) circle (0.21);
                \draw[line width=2,color=red,fill=red] (12, 2) circle (0.21);
                \draw[rounded corners=1, color=green, line width=4] (0, 0) -- (1, 1) -- (2, 0) -- (3, 1) -- (4, 0) -- (5, 1) -- (6, 2) -- (7, 3) -- (8, 2) -- (9, 1) -- (10, 0) -- (11, 1) -- (12, 2) -- (13, 1) -- (14, 0);
                \draw[dotted] (0, 0) grid (14, 4);
                \draw[rounded corners=1, color=black, line width=2] (0, 0) -- (1, 1) -- (2, 0) -- (3, 1) -- (4, 2) -- (5, 3) -- (6, 2) -- (7, 3) -- (8, 4) -- (9, 3) -- (10, 2) -- (11, 1) -- (12, 2) -- (13, 1) -- (14, 0);
              \end{tikzpicture}$}}
              sage: DyckWord([1,0])._latex_()
              '\\vcenter{\\hbox{$\\begin{tikzpicture}[scale=1]\n  \\draw[dotted] (0, 0) grid (2, 1);\n  \\draw[rounded corners=1, color=black, line width=2] (0, 0) -- (1, 1) -- (2, 0);\n\\end{tikzpicture}$}}'
              sage: DyckWord([1,0,1,1,0,0])._latex_()
              '\\vcenter{\\hbox{$\\begin{tikzpicture}[scale=1]\n  \\draw[dotted] (0, 0) grid (6, 2);\n  \\draw[rounded corners=1, color=black, line width=2] (0, 0) -- (1, 1) -- (2, 0) -- (3, 1) -- (4, 2) -- (5, 1) -- (6, 0);\n\\end{tikzpicture}$}}'
          """
          latex.add_package_to_preamble_if_available("tikz")
          heights = self.heights()
          latex_options = self.latex_options()
          diagonal = latex_options["diagonal"]
          ht = [(0, 0)]
          valleys = []
          peaks = []
          for i in range(1, len(heights)):
              a, b = ht[-1]
              if heights[i] > heights[i-1]:
                  if diagonal:
                      ht.append((a, b+1))
                  else:
                      ht.append((a+1, b+1))
                  if i < len(heights)-1 and heights[i+1] < heights[i]:
                      peaks.append(ht[-1])
              else:
                  if diagonal:
                      ht.append((a+1, b))
                  else:
                      ht.append((a+1, b-1))
                  if i < len(heights)-1 and heights[i+1] > heights[i]:
                      valleys.append(ht[-1])
          ht = iter(ht)
          if diagonal:
              grid = [((0, i), (i, i+1))
                      for i in range(self.number_of_open_symbols())]
          else:
              grid = [((0, 0), (len(self), self.height()))]
          res = "\\vcenter{\\hbox{$\\begin{tikzpicture}[scale="+str(latex_options['tikz_scale'])+"]\n"
          mark_points = []
          if latex_options['valleys']:
              mark_points.extend(valleys)
          if latex_options['peaks']:
              mark_points.extend(peaks)
          for v in mark_points:
              res += "  \\draw[line width=2,color=red,fill=red] %s circle (%s);\n" % (str(v), 0.15 + .03 * latex_options['line width'])
          if latex_options["bounce path"]:
              D = self.bounce_path()
              D.set_latex_options(latex_options)
              D.set_latex_options({"color": "green",
                                   "line width": 2 * latex_options['line width'],
                                   "bounce path": False,
                                   "peaks": False, "valleys": False})
              res += D._latex_().split("\n")[-2] + "\n"
          for v1, v2 in grid:
              res += "  \\draw[dotted] %s grid %s;\n" % (str(v1), str(v2))
          if diagonal:
              res += "  \\draw (0,0) -- %s;\n" % str((self.number_of_open_symbols(), self.number_of_open_symbols()))
          res += "  \\draw[rounded corners=1, color=%s, line width=%s] (0, 0)" % (latex_options['color'], str(latex_options['line width']))
          next(ht)
          for i, j in ht:
              res += " -- (%s, %s)" % (i, j)
          res += ";\n"
          res += "\\end{tikzpicture}$}}"
          return res
  
      def plot(self, **kwds):
          """
          Plot a Dyck word as a continuous path.
  
          EXAMPLES::
  
              sage: w = DyckWords(100).random_element()
              sage: w.plot()
              Graphics object consisting of 1 graphics primitive
          """
          from sage.plot.plot import list_plot
          step = [-1, 1]
          sigma = 0
          list_sigma = [0]
          for l in self:
              sigma += step[l]
              list_sigma.append(sigma)
          return list_plot(list_sigma, plotjoined=True, **kwds)
  
      def length(self):
          r"""
          Return the length of ``self``.
  
          EXAMPLES::
  
              sage: DyckWord([1, 0, 1, 0]).length()
              4
              sage: DyckWord([1, 0, 1, 1, 0]).length()
              5
  
          TESTS::
  
              sage: DyckWord([]).length()
              0
          """
          return len(self)
  
      def number_of_open_symbols(self):
          r"""
          Return the number of open symbols in ``self``.
  
          EXAMPLES::
  
              sage: DyckWord([1, 0, 1, 0]).number_of_open_symbols()
              2
              sage: DyckWord([1, 0, 1, 1, 0]).number_of_open_symbols()
              3
  
          TESTS::
  
              sage: DyckWord([]).number_of_open_symbols()
              0
          """
          return len([x for x in self if x == open_symbol])
  
      def number_of_close_symbols(self):
          r"""
          Return the number of close symbols in ``self``.
  
          EXAMPLES::
  
              sage: DyckWord([1, 0, 1, 0]).number_of_close_symbols()
              2
              sage: DyckWord([1, 0, 1, 1, 0]).number_of_close_symbols()
              2
  
          TESTS::
  
              sage: DyckWord([]).number_of_close_symbols()
              0
          """
          return len([x for x in self if x == close_symbol])
  
      def is_complete(self):
          r"""
          Return ``True`` if ``self`` is complete.
  
          A Dyck word `d` is complete if `d` contains as many closers as openers.
  
          EXAMPLES::
  
              sage: DyckWord([1, 0, 1, 0]).is_complete()
              True
              sage: DyckWord([1, 0, 1, 1, 0]).is_complete()
              False
  
          TESTS::
  
              sage: DyckWord([]).is_complete()
              True
          """
          return self.number_of_open_symbols() == self.number_of_close_symbols()
  
      def height(self):
          r"""
          Return the height of ``self``.
  
          We view the Dyck word as a Dyck path from `(0, 0)` to
          `(2n, 0)` in the first quadrant by letting ``1``'s represent
          steps in the direction `(1, 1)` and ``0``'s represent steps in
          the direction `(1, -1)`.
  
          The height is the maximum `y`-coordinate reached.
  
          .. SEEALSO:: :meth:`heights`
  
          EXAMPLES::
  
              sage: DyckWord([]).height()
              0
              sage: DyckWord([1,0]).height()
              1
              sage: DyckWord([1, 1, 0, 0]).height()
              2
              sage: DyckWord([1, 1, 0, 1, 0]).height()
              2
              sage: DyckWord([1, 1, 0, 0, 1, 0]).height()
              2
              sage: DyckWord([1, 0, 1, 0]).height()
              1
              sage: DyckWord([1, 1, 0, 0, 1, 1, 1, 0, 0, 0]).height()
              3
          """
          # calling max(self.heights()) has a significant overhead (20%)
          height = 0
          height_max = 0
          for letter in self:
              if letter == open_symbol:
                  height += 1
                  height_max = max(height, height_max)
              elif letter == close_symbol:
                  height -= 1
          return height_max
  
      def heights(self):
          r"""
          Return the heights of ``self``.
  
          We view the Dyck word as a Dyck path from `(0,0)` to
          `(2n,0)` in the first quadrant by letting ``1``'s represent
          steps in the direction `(1,1)` and ``0``'s represent steps in
          the direction `(1,-1)`.
  
          The heights is the sequence of the `y`-coordinates of all
          `2n+1` lattice points along the path.
  
          .. SEEALSO:: :meth:`from_heights`, :meth:`min_from_heights`
  
          EXAMPLES::
  
              sage: DyckWord([]).heights()
              (0,)
              sage: DyckWord([1,0]).heights()
              (0, 1, 0)
              sage: DyckWord([1, 1, 0, 0]).heights()
              (0, 1, 2, 1, 0)
              sage: DyckWord([1, 1, 0, 1, 0]).heights()
              (0, 1, 2, 1, 2, 1)
              sage: DyckWord([1, 1, 0, 0, 1, 0]).heights()
              (0, 1, 2, 1, 0, 1, 0)
              sage: DyckWord([1, 0, 1, 0]).heights()
              (0, 1, 0, 1, 0)
              sage: DyckWord([1, 1, 0, 0, 1, 1, 1, 0, 0, 0]).heights()
              (0, 1, 2, 1, 0, 1, 2, 3, 2, 1, 0)
          """
          height = 0
          heights = [0] * (len(self) + 1)
          for i, letter in enumerate(self):
              if letter == open_symbol:
                  height += 1
              elif letter == close_symbol:
                  height -= 1
              heights[i + 1] = height
          return tuple(heights)
  
      def associated_parenthesis(self, pos):
          r"""
          Report the position for the parenthesis in ``self`` that matches the
          one at position ``pos``.
  
          The positions in ``self`` are counted from `0`.
  
          INPUT:
  
          - ``pos`` -- the index of the parenthesis in the list
  
          OUTPUT:
  
          - Integer representing the index of the matching parenthesis.
            If no parenthesis matches, return ``None``.
  
          EXAMPLES::
  
              sage: DyckWord([1, 0]).associated_parenthesis(0)
              1
              sage: DyckWord([1, 0, 1, 0]).associated_parenthesis(0)
              1
              sage: DyckWord([1, 0, 1, 0]).associated_parenthesis(1)
              0
              sage: DyckWord([1, 0, 1, 0]).associated_parenthesis(2)
              3
              sage: DyckWord([1, 0, 1, 0]).associated_parenthesis(3)
              2
              sage: DyckWord([1, 1, 0, 0]).associated_parenthesis(0)
              3
              sage: DyckWord([1, 1, 0, 0]).associated_parenthesis(2)
              1
              sage: DyckWord([1, 1, 0]).associated_parenthesis(1)
              2
              sage: DyckWord([1, 1]).associated_parenthesis(0)
          """
          d = 0
          height = 0
          if pos >= len(self):
              raise ValueError("invalid index")
  
          if self[pos] == open_symbol:
              d += 1
              height += 1
          elif self[pos] == close_symbol:
              d -= 1
              height -= 1
          else:
              raise ValueError("unknown symbol %s" % self[pos])
  
          while height != 0:
              pos += d
              if pos < 0 or pos >= len(self):
                  return None
              if self[pos] == open_symbol:
                  height += 1
              elif self[pos] == close_symbol:
                  height -= 1
          return pos
  
      def number_of_initial_rises(self):
          r"""
          Return the length of the initial run of ``self``
  
          OUTPUT:
  
          - a non--negative integer indicating the length of the initial rise
  
          EXAMPLES::
  
              sage: DyckWord([1, 0, 1, 0]).number_of_initial_rises()
              1
              sage: DyckWord([1, 1, 0, 0]).number_of_initial_rises()
              2
              sage: DyckWord([1, 1, 0, 0, 1, 0]).number_of_initial_rises()
              2
              sage: DyckWord([1, 0, 1, 1, 0, 0]).number_of_initial_rises()
              1
  
          TESTS::
  
              sage: DyckWord([]).number_of_initial_rises()
              0
              sage: DyckWord([1, 0]).number_of_initial_rises()
              1
          """
          if not self:
              return 0
          i = 1
          while self[i] == open_symbol:
              i += 1
          return i
  
      def peaks(self):
          r"""
          Return a list of the positions of the peaks of a Dyck word.
  
          A peak is `1` followed by a `0`.  Note that this does not agree with
          the definition given in [Hag2008]_.
  
          EXAMPLES::
  
              sage: DyckWord([1, 0, 1, 0]).peaks()
              [0, 2]
              sage: DyckWord([1, 1, 0, 0]).peaks()
              [1]
              sage: DyckWord([1,1,0,1,0,1,0,0]).peaks() # Haglund's def gives 2
              [1, 3, 5]
          """
          return [i for i in range(len(self)-1)
                  if self[i] == open_symbol and self[i+1] == close_symbol]
  
      def number_of_peaks(self):
          r"""
          The number of peaks of the Dyck path associated to ``self`` .
  
          .. SEEALSO:: :meth:`peaks`
  
          EXAMPLES::
  
              sage: DyckWord([1, 0, 1, 0]).number_of_peaks()
              2
              sage: DyckWord([1, 1, 0, 0]).number_of_peaks()
              1
              sage: DyckWord([1,1,0,1,0,1,0,0]).number_of_peaks()
              3
              sage: DyckWord([]).number_of_peaks()
              0
          """
          return len(self.peaks())
  
      def valleys(self):
          r"""
          Return a list of the positions of the valleys of a Dyck word.
  
          A valley is `0` followed by a `1`.
  
          EXAMPLES::
  
              sage: DyckWord([1, 0, 1, 0]).valleys()
              [1]
              sage: DyckWord([1, 1, 0, 0]).valleys()
              []
              sage: DyckWord([1,1,0,1,0,1,0,0]).valleys()
              [2, 4]
          """
          return [i for i in xrange(len(self)-1)
                  if self[i] == close_symbol and self[i+1] == open_symbol]
  
      def number_of_valleys(self):
          r"""
          Return the number of valleys of ``self``.
  
          EXAMPLES::
  
              sage: DyckWord([1, 0, 1, 0]).number_of_valleys()
              1
              sage: DyckWord([1, 1, 0, 0]).number_of_valleys()
              0
              sage: DyckWord([1, 1, 0, 0, 1, 0]).number_of_valleys()
              1
              sage: DyckWord([1, 0, 1, 1, 0, 0]).number_of_valleys()
              1
  
          TESTS::
  
              sage: DyckWord([]).number_of_valleys()
              0
              sage: DyckWord([1, 0]).number_of_valleys()
              0
          """
          return len(self.valleys())
  
      def position_of_first_return(self):
          r"""
          Return the number of vertical steps before the Dyck path returns to
          the main diagonal.
  
          EXAMPLES::
  
              sage: DyckWord([1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0]).position_of_first_return()
              1
              sage: DyckWord([1, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0]).position_of_first_return()
              7
              sage: DyckWord([1, 1, 0, 0]).position_of_first_return()
              2
              sage: DyckWord([1, 0, 1, 0]).position_of_first_return()
              1
              sage: DyckWord([]).position_of_first_return()
              0
          """
          touches = self.touch_points()
          if touches == []:
              return 0
          else:
              return touches[0]
  
      def positions_of_double_rises(self):
          r"""
          Return a list of positions in ``self`` where there are two
          consecutive `1`'s.
  
          EXAMPLES::
  
              sage: DyckWord([1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0]).positions_of_double_rises()
              [2, 5]
              sage: DyckWord([1, 1, 0, 0]).positions_of_double_rises()
              [0]
              sage: DyckWord([1, 0, 1, 0]).positions_of_double_rises()
              []
          """
          return [i for i in xrange(len(self)-1)
                  if self[i] == self[i+1] == open_symbol]
  
      def number_of_double_rises(self):
          r"""
          Return a the number of positions in ``self`` where there are two
          consecutive `1`'s.
  
          EXAMPLES::
  
              sage: DyckWord([1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0]).number_of_double_rises()
              2
              sage: DyckWord([1, 1, 0, 0]).number_of_double_rises()
              1
              sage: DyckWord([1, 0, 1, 0]).number_of_double_rises()
              0
          """
          return len(self.positions_of_double_rises())
  
      def returns_to_zero(self):
          r"""
          Return a list of positions where ``self`` has height `0`,
          excluding the position `0`.
  
          EXAMPLES::
  
              sage: DyckWord([]).returns_to_zero()
              []
              sage: DyckWord([1, 0]).returns_to_zero()
              [2]
              sage: DyckWord([1, 0, 1, 0]).returns_to_zero()
              [2, 4]
              sage: DyckWord([1, 1, 0, 0]).returns_to_zero()
              [4]
          """
          h = self.heights()
          return [i for i in xrange(2, len(h), 2) if h[i] == 0]
  
      def touch_points(self):
          r"""
          Return the abscissae (or, equivalently, ordinates) of the
          points where the Dyck path corresponding to ``self`` (comprising
          `NE` and `SE` steps) touches the main diagonal. This includes
          the last point (if it is on the main diagonal) but excludes the
          beginning point.
  
          Note that these abscissae are precisely the entries of
          :meth:`returns_to_zero` divided by `2`.
  
          OUTPUT:
  
          - a list of integers indicating where the path touches the diagonal
  
          EXAMPLES::
  
              sage: DyckWord([1, 0, 1, 0]).touch_points()
              [1, 2]
              sage: DyckWord([1, 1, 0, 0]).touch_points()
              [2]
              sage: DyckWord([1, 1, 0, 0, 1, 0]).touch_points()
              [2, 3]
              sage: DyckWord([1, 0, 1, 1, 0, 0]).touch_points()
              [1, 3]
          """
          return [i // 2 for i in self.returns_to_zero()]
  
      def touch_composition(self):
          r"""
          Return a composition which indicates the positions where ``self``
          returns to the diagonal.
  
          This assumes ``self`` to be a complete Dyck word.
  
          OUTPUT:
  
          - a composition of length equal to the length of the Dyck word.
  
          EXAMPLES::
  
              sage: DyckWord([1, 0, 1, 0]).touch_composition()
              [1, 1]
              sage: DyckWord([1, 1, 0, 0]).touch_composition()
              [2]
              sage: DyckWord([1, 1, 0, 0, 1, 0]).touch_composition()
              [2, 1]
              sage: DyckWord([1, 0, 1, 1, 0, 0]).touch_composition()
              [1, 2]
              sage: DyckWord([]).touch_composition()
              []
          """
          from sage.combinat.composition import Composition
          if self.length() == 0:
              return Composition([])
          return Composition(descents=[i-1 for i in self.touch_points()])
  
      def number_of_touch_points(self):
          r"""
          Return the number of touches of ``self`` at the main diagonal.
  
          OUTPUT:
  
          - a non--negative integer
  
          EXAMPLES::
  
              sage: DyckWord([1, 0, 1, 0]).number_of_touch_points()
              2
              sage: DyckWord([1, 1, 0, 0]).number_of_touch_points()
              1
              sage: DyckWord([1, 1, 0, 0, 1, 0]).number_of_touch_points()
              2
              sage: DyckWord([1, 0, 1, 1, 0, 0]).number_of_touch_points()
              2
  
          TESTS::
  
              sage: DyckWord([]).number_of_touch_points()
              0
          """
          return len(self.touch_points())
  
      def rise_composition(self):
          r"""
          The sequences of lengths of runs of `1`'s in ``self``. Also equal to
          the sequence of lengths of vertical segments in the Dyck path.
  
          EXAMPLES::
  
              sage: DyckWord([1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0]).pretty_print()
                         ___
                        | x
                 _______|  .
                | x x x  . .
                | x x  . . .
               _| x  . . . .
              | x  . . . . .
              |  . . . . . .
  
              sage: DyckWord([1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0]).rise_composition()
              [2, 3, 2]
              sage: DyckWord([1,1,0,0]).rise_composition()
              [2]
              sage: DyckWord([1,0,1,0]).rise_composition()
              [1, 1]
          """
          from sage.combinat.composition import Composition
          L = list(self)
          rise_comp = []
          while L:
              i = L.index(0)
              L = L[i+1:]
              if i > 0:
                  rise_comp.append(i)
          return Composition(rise_comp)
  
      @combinatorial_map(name='to two-row standard tableau')
      def to_standard_tableau(self):
          r"""
          Return a standard tableau of shape `(a,b)` where
          `a` is the number of open symbols and `b` is the number of
          close symbols in ``self``.
  
          EXAMPLES::
  
              sage: DyckWord([]).to_standard_tableau()
              []
              sage: DyckWord([1, 0]).to_standard_tableau()
              [[1], [2]]
              sage: DyckWord([1, 1, 0, 0]).to_standard_tableau()
              [[1, 2], [3, 4]]
              sage: DyckWord([1, 0, 1, 0]).to_standard_tableau()
              [[1, 3], [2, 4]]
              sage: DyckWord([1]).to_standard_tableau()
              [[1]]
              sage: DyckWord([1, 0, 1]).to_standard_tableau()
              [[1, 3], [2]]
          """
          open_positions = []
          close_positions = []
          for i in range(len(self)):
              if self[i] == open_symbol:
                  open_positions.append(i + 1)
              else:
                  close_positions.append(i + 1)
          from sage.combinat.tableau import StandardTableau
          return StandardTableau([x for x in [open_positions, close_positions] if x != []])
  
      @combinatorial_map(name="to binary trees: up step, left tree, down step, right tree")
      def to_binary_tree(self, usemap="1L0R"):
          r"""
          Return a binary tree recursively constructed from the Dyck path
          ``self`` by the map ``usemap``. The default ``usemap`` is ``'1L0R'``
          which means:
  
          - an empty Dyck word is a leaf,
  
          - a non empty Dyck word reads `1 L 0 R` where `L` and `R` correspond
            to respectively its left and right subtrees.
  
          INPUT:
  
          - ``usemap`` -- a string, either ``'1L0R'``, ``'1R0L'``, ``'L1R0'``,
            ``'R1L0'``
  
          Other valid ``usemap`` are ``'1R0L'``, ``'L1R0'``, and ``'R1L0'``.
          These correspond to different maps from Dyck paths to binary
          trees, whose recursive definitions are hopefully clear from the
          names.
  
          EXAMPLES::
  
              sage: dw = DyckWord([1,0])
              sage: dw.to_binary_tree()
              [., .]
              sage: dw = DyckWord([])
              sage: dw.to_binary_tree()
              .
              sage: dw = DyckWord([1,0,1,1,0,0])
              sage: dw.to_binary_tree()
              [., [[., .], .]]
              sage: dw.to_binary_tree("L1R0")
              [[., .], [., .]]
              sage: dw = DyckWord([1,0,1,1,0,0,1,1,1,0,1,0,0,0])
              sage: dw.to_binary_tree() == dw.to_binary_tree("1R0L").left_right_symmetry()
              True
              sage: dw.to_binary_tree() == dw.to_binary_tree("L1R0").left_border_symmetry()
              False
              sage: dw.to_binary_tree("1R0L") == dw.to_binary_tree("L1R0").left_border_symmetry()
              True
              sage: dw.to_binary_tree("R1L0") == dw.to_binary_tree("L1R0").left_right_symmetry()
              True
              sage: dw.to_binary_tree("R10L")
              Traceback (most recent call last):
              ...
              ValueError: R10L is not a correct map
          """
          if usemap not in ["1L0R", "1R0L", "L1R0", "R1L0"]:
              raise ValueError("%s is not a correct map" % (usemap))
          from sage.combinat.binary_tree import BinaryTree
          if len(self) == 0:
              return BinaryTree()
          tp = [0]
          tp.extend(self.touch_points())
          l = len(self)
          if usemap[0] == '1':  # we check what kind of reduction we want
              s0 = 1  # start point for first substree
              e0 = tp[1] * 2 - 1  # end point for first subtree
              s1 = e0 + 1  # start point for second subtree
              e1 = l  # end point for second subtree
          else:
              s0 = 0
              e0 = tp[len(tp) - 2] * 2
              s1 = e0 + 1
              e1 = l - 1
          trees = [DyckWord(self[s0:e0]).to_binary_tree(usemap),
                   DyckWord(self[s1:e1]).to_binary_tree(usemap)]
          if usemap[0] == "R" or usemap[1] == "R":
              trees.reverse()
          return BinaryTree(trees)
  
      @combinatorial_map(name="to the Tamari corresponding Binary tree")
      def to_binary_tree_tamari(self):
          r"""
          Return the binary tree corresponding to ``self`` in a way which
          is consistent with the Tamari orders on the set of Dyck paths and
          on the set of binary trees.
  
          This is the ``'L1R0'`` map documented in :meth:`to_binary_tree`.
  
          EXAMPLES::
  
              sage: DyckWord([1,0]).to_binary_tree_tamari()
              [., .]
              sage: DyckWord([1,0,1,1,0,0]).to_binary_tree_tamari()
              [[., .], [., .]]
              sage: DyckWord([1,0,1,0,1,0]).to_binary_tree_tamari()
              [[[., .], .], .]
          """
          return self.to_binary_tree("L1R0")
  
      def tamari_interval(self, other):
          r"""
          Return the Tamari interval between ``self`` and ``other`` as a
          :class:`~sage.combinat.interval_posets.TamariIntervalPoset`.
  
          A "Tamari interval" means an interval in the Tamari order. The
          Tamari order on the set of Dyck words of size `n` is the
          partial order obtained from the Tamari order on the set of
          binary trees of size `n` (see
          :meth:`~sage.combinat.binary_tree.BinaryTree.tamari_lequal`)
          by means of the Tamari bijection between Dyck words and binary
          trees
          (:meth:`~sage.combinat.binary_tree.BinaryTree.to_dyck_word_tamari`).
  
          INPUT:
  
          - ``other`` -- a Dyck word greater or equal to ``self`` in the
            Tamari order
  
          EXAMPLES::
  
              sage: dw = DyckWord([1, 1, 0, 1, 0, 0, 1, 0])
              sage: ip = dw.tamari_interval(DyckWord([1, 1, 1, 0, 0, 1, 0, 0])); ip
              The Tamari interval of size 4 induced by relations [(2, 4), (3, 4), (3, 1), (2, 1)]
              sage: ip.lower_dyck_word()
              [1, 1, 0, 1, 0, 0, 1, 0]
              sage: ip.upper_dyck_word()
              [1, 1, 1, 0, 0, 1, 0, 0]
              sage: ip.interval_cardinality()
              4
              sage: ip.number_of_tamari_inversions()
              2
              sage: list(ip.dyck_words())
              [[1, 1, 1, 0, 0, 1, 0, 0],
               [1, 1, 1, 0, 0, 0, 1, 0],
               [1, 1, 0, 1, 0, 1, 0, 0],
               [1, 1, 0, 1, 0, 0, 1, 0]]
              sage: dw.tamari_interval(DyckWord([1,1,0,0,1,1,0,0]))
              Traceback (most recent call last):
              ...
              ValueError: The two Dyck words are not comparable on the Tamari lattice.
          """
          from sage.combinat.interval_posets import TamariIntervalPosets
          return TamariIntervalPosets.from_dyck_words(self, other)
  
      def to_area_sequence(self):
          r"""
          Return the area sequence of the Dyck word ``self``.
  
          The area sequence of a Dyck word `w` is defined as follows:
          Representing the Dyck word `w` as a Dyck path from `(0, 0)` to
          `(n, n)` using `N` and `E` steps (this involves padding `w` by
          `E` steps until `w` reaches the main diagonal if `w` is not
          already a complete Dyck path), the area sequence of `w` is the
          sequence `(a_1, a_2, \ldots, a_n)`, where `a_i` is the number
          of full cells in the `i`-th row of the rectangle
          `[0, n] \times [0, n]` which lie completely above the diagonal.
          (The cells are the regions into which the rectangle is
          subdivided by the lines `x = i` with `i` integer and the lines
          `y = j` with `j` integer. The `i`-th row consists of all the
          cells between the lines `y = i-1` and `y = i`.)
  
          An alternative definition:
          Representing the Dyck word `w` as a Dyck path consisting of
          `NE` and `SE` steps, the area sequence is the sequence of
          ordinates of all lattice points on the path which are
          starting points of `NE` steps.
  
          A list of integers `l` is the area sequence of some Dyck path
          if and only if it satisfies `l_0 = 0` and
          `0 \leq l_{i+1} \leq l_i + 1` for `i > 0`.
  
          EXAMPLES::
  
              sage: DyckWord([]).to_area_sequence()
              []
              sage: DyckWord([1, 0]).to_area_sequence()
              [0]
              sage: DyckWord([1, 1, 0, 0]).to_area_sequence()
              [0, 1]
              sage: DyckWord([1, 0, 1, 0]).to_area_sequence()
              [0, 0]
              sage: all(dw ==
              ....:     DyckWords().from_area_sequence(dw.to_area_sequence())
              ....:     for i in range(6) for dw in DyckWords(i))
              True
              sage: DyckWord([1,0,1,0,1,0,1,0,1,0]).to_area_sequence()
              [0, 0, 0, 0, 0]
              sage: DyckWord([1,1,1,1,1,0,0,0,0,0]).to_area_sequence()
              [0, 1, 2, 3, 4]
              sage: DyckWord([1,1,1,1,0,1,0,0,0,0]).to_area_sequence()
              [0, 1, 2, 3, 3]
              sage: DyckWord([1,1,0,1,0,0,1,1,0,1,0,1,0,0]).to_area_sequence()
              [0, 1, 1, 0, 1, 1, 1]
          """
          seq = []
          a = 0
          for move in self:
              if move == open_symbol:
                  seq.append(a)
                  a += 1
              else:
                  a -= 1
          return seq
  
  
  class DyckWord_complete(DyckWord):
      r"""
      The class of complete
      :class:`Dyck words<sage.combinat.dyck_word.DyckWord>`.
      A Dyck word is complete, if it contains as many closers as openers.
  
      For further information on Dyck words, see
      :class:`DyckWords_class<sage.combinat.dyck_word.DyckWord>`.
      """
      def semilength(self):
          r"""
          Return the semilength of ``self``.
  
          The semilength of a complete Dyck word `d` is the number of openers
          and the number of closers.
  
          EXAMPLES::
  
              sage: DyckWord([1, 0, 1, 0]).semilength()
              2
  
          TESTS::
  
              sage: DyckWord([]).semilength()
              0
          """
          return len(self) // 2
  
      @combinatorial_map(name='to partition')
      def to_partition(self):
          r"""
          Return the partition associated to ``self`` .
  
          This partition is determined by thinking of ``self`` as a lattice path
          and considering the cells which are above the path but within the
          `n \times n`  grid and the partition is formed by reading the sequence
          of the number of cells in this collection in each row.
  
          OUTPUT:
  
          - a partition representing the rows of cells in the square lattice
            and above the path
  
          EXAMPLES::
  
              sage: DyckWord([]).to_partition()
              []
              sage: DyckWord([1,0]).to_partition()
              []
              sage: DyckWord([1,1,0,0]).to_partition()
              []
              sage: DyckWord([1,0,1,0]).to_partition()
              [1]
              sage: DyckWord([1,0,1,0,1,0]).to_partition()
              [2, 1]
              sage: DyckWord([1,1,0,0,1,0]).to_partition()
              [2]
              sage: DyckWord([1,0,1,1,0,0]).to_partition()
              [1, 1]
          """
          from sage.combinat.partition import Partition
          n = len(self) // 2
          res = []
          for c in reversed(self):
              if c == close_symbol:
                  n -= 1
              else:
                  res.append(n)
          return Partition(res)
  
      def number_of_parking_functions(self):
          r"""
          Return the number of parking functions with ``self`` as the supporting
          Dyck path.
  
          One representation of a parking function is as a pair consisting of a
          Dyck path and a permutation `\pi` such that if
          `[a_0, a_1, \ldots, a_{n-1}]` is the area_sequence of the Dyck path
          (see :meth:`to_area_sequence<DyckWord.to_area_sequence>`) then the
          permutation `\pi` satisfies `\pi_i < \pi_{i+1}` whenever
          `a_{i} < a_{i+1}`. This function counts the number of permutations `\pi`
          which satisfy this condition.
  
          EXAMPLES::
  
              sage: DyckWord(area_sequence=[0,1,2]).number_of_parking_functions()
              1
              sage: DyckWord(area_sequence=[0,1,1]).number_of_parking_functions()
              3
              sage: DyckWord(area_sequence=[0,1,0]).number_of_parking_functions()
              3
              sage: DyckWord(area_sequence=[0,0,0]).number_of_parking_functions()
              6
          """
          from sage.arith.all import multinomial
          return multinomial(list(self.rise_composition()))
  
      def list_parking_functions(self):
          r"""
          Return all parking functions whose supporting Dyck path is ``self``.
  
          EXAMPLES::
  
              sage: DyckWord([1,1,0,0,1,0]).list_parking_functions()
              Permutations of the multi-set [1, 1, 3]
              sage: DyckWord([1,1,1,0,0,0]).list_parking_functions()
              Permutations of the multi-set [1, 1, 1]
              sage: DyckWord([1,0,1,0,1,0]).list_parking_functions()
              Permutations of the set [1, 2, 3]
          """
          alist = self.to_area_sequence()
          return Permutations([i - alist[i]+1 for i in range(len(alist))])
          # TODO: upon implementation of ParkingFunction class
          # map(ParkingFunction, Permutations([i - alist[i]+1 for i in range(len(alist))]))
  
      def reading_permutation(self):
          r"""
          The permutation formed by taking the reading word of the Dyck path
          representing ``self`` (with `N` and `E` steps) if the vertical
          edges of the Dyck path are labeled from bottom to top with `1`
          through `n` and the diagonals are read from top to bottom starting
          with the diagonal furthest from the main diagonal.
  
          EXAMPLES::
  
              sage: DyckWord([1,0,1,0]).reading_permutation()
              [2, 1]
              sage: DyckWord([1,1,0,0]).reading_permutation()
              [2, 1]
              sage: DyckWord([1,1,0,1,0,0]).reading_permutation()
              [3, 2, 1]
              sage: DyckWord([1,1,0,0,1,0]).reading_permutation()
              [2, 3, 1]
              sage: DyckWord([1,0,1,1,0,0,1,0]).reading_permutation()
              [3, 4, 2, 1]
          """
          alist = self.to_area_sequence()
          if len(alist) == 0:
              return Permutation([])
          m = max(alist)
          p1 = Word([m-alist[-i-1]
                     for i in range(len(alist))]).standard_permutation()
          return p1.inverse().complement()
  
      def characteristic_symmetric_function(self, q=None,
                                            R=QQ['q', 't'].fraction_field()):
          r"""
          The characteristic function of ``self`` is the sum of
          `q^{dinv(D,F)} Q_{ides(read(D,F))}` over all permutation
          fillings of the Dyck path representing ``self``, where
          `ides(read(D,F))` is the descent composition of the inverse of the
          reading word of the filling.
  
          INPUT:
  
          - ``q`` -- (default: ``q = R('q')``) a parameter for the generating
            function power
  
          - ``R`` -- (default : ``R = QQ['q','t'].fraction_field()``) the base
            ring to do the calculations over
  
          OUTPUT:
  
          - an element of the symmetric functions over the ring ``R``
            (in the Schur basis).
  
          EXAMPLES::
  
              sage: R = QQ['q','t'].fraction_field()
              sage: (q,t) = R.gens()
              sage: f = sum(t**D.area()*D.characteristic_symmetric_function() for D in DyckWords(3)); f
              (q^3+q^2*t+q*t^2+t^3+q*t)*s[1, 1, 1] + (q^2+q*t+t^2+q+t)*s[2, 1] + s[3]
              sage: f.nabla(power=-1)
              s[1, 1, 1]
          """
          from sage.combinat.ncsf_qsym.qsym import QuasiSymmetricFunctions
          from sage.combinat.sf.sf import SymmetricFunctions
          if q is None:
              q = R('q')
          else:
              if not q in R:
                  raise ValueError("q=%s must be an element of the base ring %s" % (q, R))
          F = QuasiSymmetricFunctions(R).Fundamental()
          p = self.reading_permutation().inverse()
          perms = [Word(perm).standard_permutation()
                   for perm in self.list_parking_functions()]
          QSexpr = sum(q**self.dinv(pv.inverse())*F(Permutation([p(i) for i in pv]).descents_composition()) for pv in perms)
          s = SymmetricFunctions(R).s()
          return s(QSexpr.to_symmetric_function())
  
      def to_pair_of_standard_tableaux(self):
          r"""
          Convert ``self`` to a pair of standard tableaux of the same shape and
          of length less than or equal to two.
  
          EXAMPLES::
  
              sage: DyckWord([1,0,1,0]).to_pair_of_standard_tableaux()
              ([[1], [2]], [[1], [2]])
              sage: DyckWord([1,1,0,0]).to_pair_of_standard_tableaux()
              ([[1, 2]], [[1, 2]])
              sage: DyckWord([1,1,0,1,0,0,1,1,0,1,0,1,0,0]).to_pair_of_standard_tableaux()
              ([[1, 2, 4, 7], [3, 5, 6]], [[1, 2, 4, 6], [3, 5, 7]])
          """
          from sage.combinat.tableau import Tableau
          n = self.semilength()
          if n == 0:
              return (Tableau([]), Tableau([]))
          elif self.height() == n:
              return (Tableau([range(1, n + 1)]), Tableau([range(1, n + 1)]))
          else:
              left = [[], []]
              right = [[], []]
              for pos in range(n):
                  if self[pos] == open_symbol:
                      left[0].append(pos + 1)
                  else:
                      left[1].append(pos + 1)
                  if self[-pos-1] == close_symbol:
                      right[0].append(pos+1)
                  else:
                      right[1].append(pos+1)
              return (Tableau(left), Tableau(right))
  
      @combinatorial_map(name='to 312 avoiding permutation')
      def to_312_avoiding_permutation(self):
          r"""
          Convert ``self`` to a `312`-avoiding permutation using the bijection by
          Bandlow and Killpatrick in [BK2001]_.  Sends the area to the
          inversion number.
  
          REFERENCES:
  
          .. [BK2001] \J. Bandlow, K. Killpatrick -- An area-to_inv bijection
             between Dyck paths and 312-avoiding permutations, Electronic Journal
             of Combinatorics, Volume 8, Issue 1 (2001).
  
          EXAMPLES::
  
              sage: DyckWord([1,1,0,0]).to_312_avoiding_permutation()
              [2, 1]
              sage: DyckWord([1,0,1,0]).to_312_avoiding_permutation()
              [1, 2]
              sage: p = DyckWord([1,1,0,1,0,0,1,1,0,1,0,1,0,0]).to_312_avoiding_permutation(); p
              [2, 3, 1, 5, 6, 7, 4]
              sage: DyckWord([1,1,0,1,0,0,1,1,0,1,0,1,0,0]).area()
              5
              sage: p.length()
              5
  
          TESTS::
  
              sage: PD = [D.to_312_avoiding_permutation() for D in DyckWords(5)]
              sage: all(pi.avoids([3,1,2]) for pi in PD)
              True
              sage: all(D.area()==D.to_312_avoiding_permutation().length() for D in DyckWords(5))
              True
          """
          n = self.semilength()
          area = self.to_area_sequence()
          from sage.groups.perm_gps.permgroup_named import SymmetricGroup
          pi = SymmetricGroup(n).one()
          for j in range(n):
              for i in range(area[j]):
                  pi = pi.apply_simple_reflection(j-i)
          return Permutation(~pi)
  
      @combinatorial_map(name='to non-crossing permutation')
      def to_noncrossing_permutation(self):
          r"""
          Use the bijection by C. Stump in [Stu2008]_ to send ``self`` to a
          non-crossing permutation.
  
          A non-crossing permutation when written in cyclic notation has cycles
          which are strictly increasing.  Sends the area to the inversion number
          and ``self.major_index()`` to `n(n-1) - maj(\sigma) - maj(\sigma^{-1})`.
          Uses the function :func:`~sage.combinat.dyck_word.pealing`
  
          REFERENCES:
  
          .. [Stu2008] \C. Stump -- More bijective Catalan combinatorics on
             permutations and on colored permutations, Preprint.
             :arXiv:`0808.2822`.
  
          EXAMPLES::
  
              sage: DyckWord([1,1,0,0]).to_noncrossing_permutation()
              [2, 1]
              sage: DyckWord([1,0,1,0]).to_noncrossing_permutation()
              [1, 2]
              sage: p = DyckWord([1,1,0,1,0,0,1,1,0,1,0,1,0,0]).to_noncrossing_permutation(); p
              [2, 3, 1, 5, 6, 7, 4]
              sage: DyckWord([1,1,0,1,0,0,1,1,0,1,0,1,0,0]).area()
              5
              sage: p.length()
              5
  
          TESTS::
  
              sage: all(D.area()==D.to_noncrossing_permutation().length() for D in DyckWords(5))
              True
              sage: all(20-D.major_index()==D.to_noncrossing_permutation().major_index()
              ....:     +D.to_noncrossing_permutation().imajor_index() for D in DyckWords(5))
              True
          """
          n = self.semilength()
          if n == 0:
              return Permutation([])
          D, touch_sequence = pealing(self, return_touches=True)
          from sage.groups.perm_gps.permgroup_named import SymmetricGroup
          S = SymmetricGroup(n)
          pi = S.one()
          while touch_sequence:
              for touches in touch_sequence:
                  pi = pi * S(tuple(touches))
              D, touch_sequence = pealing(D, return_touches=True)
          return Permutation(pi)
  
      @combinatorial_map(name='to 321 avoiding permutation')
      def to_321_avoiding_permutation(self):
          r"""
          Use the bijection (pp. 60-61 of [Knu1973]_ or section 3.1 of [CK2008]_)
          to send ``self`` to a `321`-avoiding permutation.
  
          It is shown in [EP2004]_ that it sends the number of centered tunnels
          to the number of fixed points, the number of right tunnels to the
          number of exceedences, and the semilength plus the height of the middle
          point to 2 times the length of the longest increasing subsequence.
  
          REFERENCES:
  
          .. [EP2004] \S. Elizalde, I. Pak. *Bijections for refined restricted
             permutations**. JCTA 105(2) 2004.
          .. [CK2008] \A. Claesson, S. Kitaev. *Classification of bijections
             between `321`- and `132`- avoiding permutations*. Seminaire
             Lotharingien de Combinatoire **60** 2008. :arxiv:`0805.1325`.
          .. [Knu1973] \D. Knuth. *The Art of Computer Programming, Vol. III*.
             Addison-Wesley. Reading, MA. 1973.
  
          EXAMPLES::
  
              sage: DyckWord([1,0,1,0]).to_321_avoiding_permutation()
              [2, 1]
              sage: DyckWord([1,1,0,0]).to_321_avoiding_permutation()
              [1, 2]
              sage: D = DyckWord([1,1,0,1,0,0,1,1,0,1,0,1,0,0])
              sage: p = D.to_321_avoiding_permutation()
              sage: p
              [3, 5, 1, 6, 2, 7, 4]
              sage: D.number_of_tunnels()
              0
              sage: p.number_of_fixed_points()
              0
              sage: D.number_of_tunnels('right')
              4
              sage: len(p.weak_excedences())-p.number_of_fixed_points()
              4
              sage: n = D.semilength()
              sage: D.heights()[n] + n
              8
              sage: 2*p.longest_increasing_subsequence_length()
              8
  
          TESTS::
  
              sage: PD = [D.to_321_avoiding_permutation() for D in DyckWords(5)]
              sage: all(pi.avoids([3,2,1]) for pi in PD)
              True
              sage: to_perm = lambda x: x.to_321_avoiding_permutation()
              sage: all(D.number_of_tunnels() == to_perm(D).number_of_fixed_points()
              ....:     for D in DyckWords(5))
              True
              sage: all(D.number_of_tunnels('right') == len(to_perm(D).weak_excedences())
              ....:     -to_perm(D).number_of_fixed_points() for D in DyckWords(5))
              True
              sage: all(D.heights()[5]+5 == 2*to_perm(D).longest_increasing_subsequence_length()
              ....:     for D in DyckWords(5))
              True
          """
          from sage.combinat.rsk import RSK_inverse
          A, B = self.to_pair_of_standard_tableaux()
          return RSK_inverse(A, B, output='permutation')
  
      @combinatorial_map(name='to 132 avoiding permutation')
      def to_132_avoiding_permutation(self):
          r"""
          Use the bijection by C. Krattenthaler in [Kra2001]_ to send ``self``
          to a `132`-avoiding permutation.
  
          REFERENCES:
  
          .. [Kra2001] \C. Krattenthaler -- Permutations with restricted
             patterns and Dyck paths, Adv. Appl. Math. 27 (2001), 510--530.
  
          EXAMPLES::
  
              sage: DyckWord([1,1,0,0]).to_132_avoiding_permutation()
              [1, 2]
              sage: DyckWord([1,0,1,0]).to_132_avoiding_permutation()
              [2, 1]
              sage: DyckWord([1,1,0,1,0,0,1,1,0,1,0,1,0,0]).to_132_avoiding_permutation()
              [6, 5, 4, 7, 2, 1, 3]
  
          TESTS::
  
              sage: PD = [D.to_132_avoiding_permutation() for D in DyckWords(5)]
              sage: all(pi.avoids([1,3,2]) for pi in PD)
              True
          """
          n = self.semilength()
          area = self.to_area_sequence()
          area.append(0)
          pi = []
          values = range(1, n+1)
          for i in range(n):
              if area[n-i-1]+1 > area[n-i]:
                  pi.append(n-i-area[n-i-1])
                  values.remove(n-i-area[n-i-1])
              else:
                  v = min(v for v in values if v > n-i-area[n-i-1])
                  pi.append(v)
                  values.remove(v)
          return Permutation(pi)
  
      def to_permutation(self, map):
          r"""
          This is simply a method collecting all implemented maps from Dyck
          words to permutations.
  
          INPUT:
  
          - ``map`` -- defines the map from Dyck words to permutations.
            These are currently:
  
            - ``Bandlow-Killpatrick``: :func:`to_312_avoiding_permutation`
            - ``Knuth``: :func:`to_321_avoiding_permutation`
            - ``Krattenthaler``: :func:`to_132_avoiding_permutation`
            - ``Stump``: :func:`to_noncrossing_permutation`
  
          EXAMPLES::
  
              sage: D = DyckWord([1,1,1,0,1,0,0,0])
              sage: D.pretty_print()
                 _____
               _| x x
              | x x  .
              | x  . .
              |  . . .
  
              sage: D.to_permutation(map="Bandlow-Killpatrick")
              [3, 4, 2, 1]
              sage: D.to_permutation(map="Stump")
              [4, 2, 3, 1]
              sage: D.to_permutation(map="Knuth")
              [1, 2, 4, 3]
              sage: D.to_permutation(map="Krattenthaler")
              [2, 1, 3, 4]
          """
          if map == "Bandlow-Killpatrick":
              return self.to_312_avoiding_permutation()
          elif map == "Knuth":
              return self.to_321_avoiding_permutation()
          elif map == "Krattenthaler":
              return self.to_132_avoiding_permutation()
          elif map == "Stump":
              return self.to_noncrossing_permutation()
          else:
              raise ValueError("The given map is not valid.")
  
      def to_noncrossing_partition(self, bijection=None):
          r"""
          Bijection of Biane from ``self`` to a noncrossing partition.
  
          There is an optional parameter ``bijection`` that indicates if a
          different bijection from Dyck words to non-crossing partitions
          should be used (since there are potentially many).
  
          If the parameter ``bijection`` is "Stump" then the bijection used is
          from [Stu2008]_, see also the method :meth:`to_noncrossing_permutation`.
  
          Thanks to Mathieu Dutour for describing the bijection.  See also
          :func:`from_noncrossing_partition`.
  
          EXAMPLES::
  
              sage: DyckWord([]).to_noncrossing_partition()
              []
              sage: DyckWord([1, 0]).to_noncrossing_partition()
              [[1]]
              sage: DyckWord([1, 1, 0, 0]).to_noncrossing_partition()
              [[1, 2]]
              sage: DyckWord([1, 1, 1, 0, 0, 0]).to_noncrossing_partition()
              [[1, 2, 3]]
              sage: DyckWord([1, 0, 1, 0, 1, 0]).to_noncrossing_partition()
              [[1], [2], [3]]
              sage: DyckWord([1, 1, 0, 1, 0, 0]).to_noncrossing_partition()
              [[2], [1, 3]]
              sage: DyckWord([]).to_noncrossing_partition("Stump")
              []
              sage: DyckWord([1, 0]).to_noncrossing_partition("Stump")
              [[1]]
              sage: DyckWord([1, 1, 0, 0]).to_noncrossing_partition("Stump")
              [[1, 2]]
              sage: DyckWord([1, 1, 1, 0, 0, 0]).to_noncrossing_partition("Stump")
              [[1, 3], [2]]
              sage: DyckWord([1, 0, 1, 0, 1, 0]).to_noncrossing_partition("Stump")
              [[1], [2], [3]]
              sage: DyckWord([1, 1, 0, 1, 0, 0]).to_noncrossing_partition("Stump")
              [[1, 2, 3]]
          """
          if bijection == "Stump":
              return [[v for v in c]
                      for c in self.to_noncrossing_permutation().cycle_tuples()]
          partition = []
          stack = []
          i = 0
          p = 1
  
          #Invariants:
          # - self[i] = 0
          # - p is the number of opening parens at position i
  
          while i < len(self):
              stack.append(p)
              j = i + 1
              while j < len(self) and self[j] == close_symbol:
                  j += 1
  
              #Now j points to the next 1 or past the end of self
              nz = j - (i + 1)  # the number of )'s between i and j
              if nz > 0:
                  # Remove the nz last elements of stack and
                  # make a new part in partition
                  if nz > len(stack):
                      raise ValueError("incorrect Dyck word")
  
                  partition.append(stack[-nz:])
  
                  stack = stack[: -nz]
              i = j
              p += 1
  
          if len(stack) > 0:
              raise ValueError("incorrect Dyck word")
  
          return partition
  
      def to_Catalan_code(self):
          r"""
          Return the Catalan code associated to ``self``.
  
          A Catalan code of length `n` is a sequence
          `(a_1, a_2, \ldots, a_n)` of `n` integers `a_i` such that:
  
          - `0 \leq a_i \leq n-i` for every `i`;
  
          - if `i < j` and `a_i > 0` and `a_j > 0` and
            `a_{i+1} = a_{i+2} = \cdots = a_{j-1} = 0`,
            then `a_i - a_j < j-i`.
  
          It turns out that the Catalan codes of length `n` are in
          bijection with Dyck words.
  
          The Catalan code of a Dyck word is example (x) in Richard Stanley's
          exercises on combinatorial interpretations for Catalan objects.
          The code in this example is the reverse of the description provided
          there. See [Sta-EC2]_ and [StaCat98]_.
  
          EXAMPLES::
  
              sage: DyckWord([]).to_Catalan_code()
              []
              sage: DyckWord([1, 0]).to_Catalan_code()
              [0]
              sage: DyckWord([1, 1, 0, 0]).to_Catalan_code()
              [0, 1]
              sage: DyckWord([1, 0, 1, 0]).to_Catalan_code()
              [0, 0]
              sage: all(dw ==
              ....:     DyckWords().from_Catalan_code(dw.to_Catalan_code())
              ....:     for i in range(6) for dw in DyckWords(i))
              True
          """
          if not self:
              return []
          cut = self.associated_parenthesis(0)
          recdw = DyckWord(self[1:cut]+self[cut+1:])
          returns = [0]+recdw.returns_to_zero()
          res = recdw.to_Catalan_code()
          res.append(returns.index(cut-1))
          return res
  
      @combinatorial_map(name="To Ordered tree")
      def to_ordered_tree(self):
          r"""
          Return the ordered tree corresponding to ``self`` where the depth
          of the tree is the maximal height of ``self``.
  
          EXAMPLES::
  
              sage: D = DyckWord([1,1,0,0])
              sage: D.to_ordered_tree()
              [[[]]]
              sage: D = DyckWord([1,0,1,0])
              sage: D.to_ordered_tree()
              [[], []]
              sage: D = DyckWord([1, 0, 1, 1, 0, 0])
              sage: D.to_ordered_tree()
              [[], [[]]]
              sage: D = DyckWord([1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0])
              sage: D.to_ordered_tree()
              [[], [[], []], [[], [[]]]]
  
          TESTS::
  
              sage: D = DyckWord([1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0])
              sage: D == D.to_ordered_tree().to_dyck_word()
              True
          """
          from sage.combinat.ordered_tree import OrderedTree
          levels = [OrderedTree().clone()]
          for u in self:
              if u == 1:
                  levels.append(OrderedTree().clone())
              else:
                  tree = levels.pop()
                  tree.set_immutable()
                  root = levels.pop()
                  root.append(tree)
                  levels.append(root)
          root = levels[0]
          root.set_immutable()
          return root
  
      def to_triangulation(self):
          r"""
          Map ``self`` to a triangulation.
  
          The map from complete Dyck words of length `2n` to
          triangulations of `n+2`-gon given by this function is a
          bijection that can be described as follows.
  
          Consider the Dyck word as a path from `(0, 0)` to `(n, n)`
          staying above the diagonal, where `1` is an up step and `0` is
          a right step. Then each horizontal step has a co-height (`0`
          at the top and `n-1` at most at the bottom). One reads the
          Dyck word from left to right. At the begining, all vertices
          from `0` to `n+1` are available. For each horizontal step,
          one creates an edge from the vertex indexed by the co-height
          to the next available vertex. This chops out a triangle from
          the polygon and one removes the middle vertex of this triangle
          from the list of available vertices.
  
          This bijection has the property that the set of smallest
          vertices of the edges in a triangulation is an encoding of the
          co-heights, from which the Dyck word can be easily recovered.
  
          OUTPUT:
  
          a list of pairs `(i, j)` that are the edges of the
          triangulations.
  
          EXAMPLES::
  
              sage: DyckWord([1, 1, 0, 0]).to_triangulation()
              [(0, 2)]
  
              sage: [t.to_triangulation() for t in DyckWords(3)]
              [[(2, 4), (1, 4)],
              [(2, 4), (0, 2)],
              [(1, 3), (1, 4)],
              [(1, 3), (0, 3)],
              [(0, 2), (0, 3)]]
  
          REFERENCES:
  
          .. [Cha2005] \F. Chapoton, Une Base Symétrique de l'algèbre des
             Coinvariants Quasi-Symétriques, Electronic Journal of
             Combinatorics Vol 12(1) (2005) N16.
          """
          n = self.number_of_open_symbols()
          l = range(n + 2)  # from 0 to n + 1
          edges = []
          coheight = n - 1
          for letter in self[1:-1]:
              if letter == 1:
                  coheight -= 1
              else:
                  edges.append((coheight, l[coheight + 2]))
                  l.pop(coheight + 1)
  
          return edges
  
      def to_triangulation_as_graph(self):
          r"""
          Map ``self`` to a triangulation and return the result as a graph.
  
          See :meth:`to_triangulation` for the bijection used to map
          complete Dyck words to triangulations.
  
          OUTPUT:
  
          - a graph containing both the perimeter edges and the inner
            edges of a triangulation of a regular polygon.
  
          EXAMPLES::
  
              sage: g = DyckWord([1, 1, 0, 0, 1, 0]).to_triangulation_as_graph()
              sage: g
              Graph on 5 vertices
              sage: g.edges(labels=False)
              [(0, 1), (0, 4), (1, 2), (1, 3), (1, 4), (2, 3), (3, 4)]
              sage: g.show()        # not tested
          """
          n = self.number_of_open_symbols()
          edges = self.to_triangulation()
          from sage.graphs.graph import Graph
          peri = [(i, i + 1) for i in range(n + 1)] + [(n + 1, 0)]
          g = Graph(n + 2)
          g.add_edges(peri)
          g.add_edges(edges)
          g.set_pos(g.layout_circular())
          return g
  
      def to_non_decreasing_parking_function(self):
          r"""
          Bijection to :class:`non-decreasing parking
          functions<sage.combinat.non_decreasing_parking_function.NonDecreasingParkingFunctions>`.
          See there the method
          :meth:`~sage.combinat.non_decreasing_parking_function.NonDecreasingParkingFunction.to_dyck_word`
          for more information.
  
          EXAMPLES::
  
              sage: DyckWord([]).to_non_decreasing_parking_function()
              []
              sage: DyckWord([1,0]).to_non_decreasing_parking_function()
              [1]
              sage: DyckWord([1,1,0,0]).to_non_decreasing_parking_function()
              [1, 1]
              sage: DyckWord([1,0,1,0]).to_non_decreasing_parking_function()
              [1, 2]
              sage: DyckWord([1,0,1,1,0,1,0,0,1,0]).to_non_decreasing_parking_function()
              [1, 2, 2, 3, 5]
  
          TESTS::
  
              sage: ld=DyckWords(5);
              sage: list(ld) == [dw.to_non_decreasing_parking_function().to_dyck_word() for dw in ld]
              True
          """
          from sage.combinat.non_decreasing_parking_function import NonDecreasingParkingFunction
          return NonDecreasingParkingFunction.from_dyck_word(self)
  
      def major_index(self):
          r"""
          Return the major index of ``self`` .
  
          The major index of a Dyck word `D` is the sum of the positions of
          the valleys of `D` (when started counting at position ``1``).
  
          EXAMPLES::
  
              sage: DyckWord([1, 0, 1, 0]).major_index()
              2
              sage: DyckWord([1, 1, 0, 0]).major_index()
              0
              sage: DyckWord([1, 1, 0, 0, 1, 0]).major_index()
              4
              sage: DyckWord([1, 0, 1, 1, 0, 0]).major_index()
              2
  
          TESTS::
  
              sage: DyckWord([]).major_index()
              0
              sage: DyckWord([1, 0]).major_index()
              0
          """
          valleys = self.valleys()
          return sum(valleys) + len(valleys)
  
      def pyramid_weight(self):
          r"""
          A pyramid of ``self`` is a subsequence of the form
          `1^h 0^h`. A pyramid is maximal if it is neither preceded by a `1`
          nor followed by a `0`.
  
          The pyramid weight of a Dyck path is the sum of the lengths of the
          maximal pyramids and was defined in [DS1992]_.
  
          EXAMPLES::
  
              sage: DyckWord([1,1,0,1,1,1,0,0,1,0,0,0,1,1,0,0]).pyramid_weight()
              6
              sage: DyckWord([1,1,1,0,0,0]).pyramid_weight()
              3
              sage: DyckWord([1,0,1,0,1,0]).pyramid_weight()
              3
              sage: DyckWord([1,1,0,1,0,0]).pyramid_weight()
              2
  
          REFERENCES:
  
          .. [DS1992] \A. Denise, R. Simion, Two combinatorial statistics on
             Dyck paths, Discrete Math 137 (1992), 155--176.
          """
          aseq = self.to_area_sequence() + [0]
          bseq = self.reverse().to_area_sequence() + [0]
          apeak = []
          bpeak = []
          for i in range(len(aseq) - 1):
              if aseq[i + 1] <= aseq[i]:
                  apeak.append(i)
              if bseq[i + 1] <= bseq[i]:
                  bpeak.append(i)
          out = 0
          for i in range(len(apeak)):
              out += min(aseq[apeak[i]]-aseq[apeak[i]+1]+1,
                         bseq[bpeak[-i-1]]-bseq[bpeak[-i-1]+1]+1)
          return out
  
      def tunnels(self):
          r"""
          Return the list of ranges of the matching parentheses in the Dyck
          word ``self``.
          That is, if ``(a,b)`` is in ``self.tunnels()``, then the matching
          parenthesis to ``self[a]`` is ``self[b-1]``.
  
          EXAMPLES::
  
              sage: DyckWord([1, 1, 0, 1, 1, 0, 0, 1, 0, 0]).tunnels()
              [(0, 10), (1, 3), (3, 7), (4, 6), (7, 9)]
          """
          heights = self.heights()
          tunnels = []
          for i in range(len(heights)-1):
              height = heights[i]
              if height < heights[i+1]:
                  tunnels.append((i, i+1+heights[i+1:].index(height)))
          return tunnels
  
      def number_of_tunnels(self, tunnel_type='centered'):
          r"""
          Return the number of tunnels of ``self`` of type ``tunnel_type``.
  
          A tunnel is a pair `(a,b)` where ``a`` is the position of an open
          parenthesis and ``b`` is the position of the matching close
          parenthesis. If `a + b = n` then the tunnel is called *centered* . If
          `a + b < n` then the tunnel is called *left* and if `a + b > n`, then
          the tunnel is called *right*.
  
          INPUT:
  
          - ``tunnel_type`` -- (default: ``'centered'``) can be one of the
            following: ``'left'``, ``'right'``, ``'centered'``, or ``'all'``.
  
          EXAMPLES::
  
              sage: DyckWord([1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0]).number_of_tunnels()
              0
              sage: DyckWord([1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0]).number_of_tunnels('left')
              5
              sage: DyckWord([1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0]).number_of_tunnels('right')
              2
              sage: DyckWord([1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0]).number_of_tunnels('all')
              7
              sage: DyckWord([1, 1, 0, 0]).number_of_tunnels('centered')
              2
          """
          n = len(self)
          tunnels = self.tunnels()
          if tunnel_type == 'left':
              return len([i for (i, j) in tunnels if i + j < n])
          elif tunnel_type == 'centered':
              return len([i for (i, j) in tunnels if i + j == n])
          elif tunnel_type == 'right':
              return len([i for (i, j) in tunnels if i + j > n])
          elif tunnel_type == 'all':
              return len(tunnels)
          else:
              raise ValueError("The given tunnel_type is not valid.")
  
      @combinatorial_map(order=2, name="Reverse path")
      def reverse(self):
          r"""
          Return the reverse and complement of ``self``.
  
          This operation corresponds to flipping the Dyck path across the
          `y=-x` line.
  
          EXAMPLES::
  
              sage: DyckWord([1,1,0,0,1,0]).reverse()
              [1, 0, 1, 1, 0, 0]
              sage: DyckWord([1,1,1,0,0,0]).reverse()
              [1, 1, 1, 0, 0, 0]
              sage: len(filter(lambda D: D.reverse() == D, DyckWords(5)))
              10
  
          TESTS::
  
              sage: DyckWord([]).reverse()
              []
          """
          list = []
          for i in range(len(self)):
              if self[i] == open_symbol:
                  list.append(close_symbol)
              else:
                  list.append(open_symbol)
          list.reverse()
          return DyckWord(list)
  
      def first_return_decomposition(self):
          r"""
          Decompose a Dyck word into a pair of Dyck words (potentially empty)
          where the first word consists of the word after the first up step and
          the corresponding matching closing parenthesis.
  
          EXAMPLES::
  
              sage: DyckWord([1,1,0,1,0,0,1,1,0,1,0,1,0,0]).first_return_decomposition()
              ([1, 0, 1, 0], [1, 1, 0, 1, 0, 1, 0, 0])
              sage: DyckWord([1,1,0,0]).first_return_decomposition()
              ([1, 0], [])
              sage: DyckWord([1,0,1,0]).first_return_decomposition()
              ([], [1, 0])
          """
          k = self.position_of_first_return() * 2
          return DyckWord(self[1:k - 1]), DyckWord(self[k:])
  
      def decomposition_reverse(self):
          r"""
          Return the involution of ``self`` with a recursive definition.
  
          If a Dyck word `D` decomposes as `1 D_1 0 D_2` where `D_1` and
          `D_2` are complete Dyck words then the decomposition reverse is
          `1 \phi(D_2) 0 \phi(D_1)`.
  
          EXAMPLES::
  
              sage: DyckWord([1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0]).decomposition_reverse()
              [1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0]
              sage: DyckWord([1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0]).decomposition_reverse()
              [1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0]
              sage: DyckWord([1,1,0,0]).decomposition_reverse()
              [1, 0, 1, 0]
              sage: DyckWord([1,0,1,0]).decomposition_reverse()
              [1, 1, 0, 0]
          """
          if self.semilength() == 0:
              return self
          else:
              D1, D2 = self.first_return_decomposition()
              return DyckWord([1] + list(D2.decomposition_reverse())
                              + [0] + list(D1.decomposition_reverse()))
  
      @combinatorial_map(name="Area-dinv to bounce-area")
      def area_dinv_to_bounce_area_map(self):
          r"""
          Return the image of ``self`` under the map which sends a
          Dyck word with ``area`` equal to `r` and ``dinv`` equal to `s` to a
          Dyck word with ``bounce`` equal to `r` and ``area`` equal to `s` .
  
          The inverse of this map is :meth:`bounce_area_to_area_dinv_map`.
  
          For a definition of this map, see [Hag2008]_ p. 50 where it is called
          `\zeta`. However, this map differs from Haglund's map by an application
          of :meth:`reverse` (as does the definition of the :meth:`bounce`
          statistic).
  
          EXAMPLES::
  
              sage: DyckWord([1,1,0,1,0,0,1,1,0,1,0,1,0,0]).area_dinv_to_bounce_area_map()
              [1, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0]
              sage: DyckWord([1,1,0,1,0,0,1,1,0,1,0,1,0,0]).area()
              5
              sage: DyckWord([1,1,0,1,0,0,1,1,0,1,0,1,0,0]).dinv()
              13
              sage: DyckWord([1,1,1,1,1,0,0,0,1,0,0,1,0,0]).area()
              13
              sage: DyckWord([1,1,1,1,1,0,0,0,1,0,0,1,0,0]).bounce()
              5
              sage: DyckWord([1,1,1,1,1,0,0,0,1,0,0,1,0,0]).area_dinv_to_bounce_area_map()
              [1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0]
              sage: DyckWord([1,1,0,0]).area_dinv_to_bounce_area_map()
              [1, 0, 1, 0]
              sage: DyckWord([1,0,1,0]).area_dinv_to_bounce_area_map()
              [1, 1, 0, 0]
          """
          a = self.to_area_sequence()
          if a == []:
              return self
          a.reverse()
          image = []
          for i in range(max(a), -2, -1):
              for j in a:
                  if j == i:
                      image.append(1)
                  elif j == i + 1:
                      image.append(0)
          return DyckWord(image)
  
      @combinatorial_map(name="Bounce-area to area-dinv")
      def bounce_area_to_area_dinv_map(D):
          r"""
          Return the image of the Dyck word under the map which sends a
          Dyck word with ``bounce`` equal to `r` and ``area`` equal to `s` to a
          Dyck word with ``area`` equal to `r` and ``dinv`` equal to `s` .
  
          This implementation uses a recursive method by saying that
          the last entry in the area sequence of `D` is equal to the number of
          touch points of the Dyck path minus 1 of the image of this map.
  
          The inverse of this map is :meth:`area_dinv_to_bounce_area_map`.
  
          For a definition of this map, see [Hag2008]_ p. 50 where it is called
          `\zeta^{-1}`. However, this map differs from Haglund's map by an
          application of :meth:`reverse` (as does the definition of the
          :meth:`bounce` statistic).
  
          EXAMPLES::
  
              sage: DyckWord([1,1,0,1,0,0,1,1,0,1,0,1,0,0]).bounce_area_to_area_dinv_map()
              [1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 0]
              sage: DyckWord([1,1,0,1,0,0,1,1,0,1,0,1,0,0]).area()
              5
              sage: DyckWord([1,1,0,1,0,0,1,1,0,1,0,1,0,0]).bounce()
              9
              sage: DyckWord([1,1,0,0,1,1,1,1,0,0,1,0,0,0]).area()
              9
              sage: DyckWord([1,1,0,0,1,1,1,1,0,0,1,0,0,0]).dinv()
              5
              sage: all(D==D.bounce_area_to_area_dinv_map().area_dinv_to_bounce_area_map() for D in DyckWords(6))
              True
              sage: DyckWord([1,1,0,0]).bounce_area_to_area_dinv_map()
              [1, 0, 1, 0]
              sage: DyckWord([1,0,1,0]).bounce_area_to_area_dinv_map()
              [1, 1, 0, 0]
          """
          aseq = D.to_area_sequence()
          out = []
          zeros = []
          for i in range(len(aseq)):
              p = (zeros+[len(out)])[aseq[i]]
              out = [1] + out[p:]+[0] + out[:p]
              zeros = [0] + [j+len(out)-p for j in zeros[:aseq[i]]]
          return DyckWord(out)
  
      def area(self):
          r"""
          Return the area for ``self`` corresponding to the area
          of the Dyck path.
  
          One can view a balanced Dyck word as a lattice path from
          `(0,0)` to `(n,n)` in the first quadrant by letting
          '1's represent steps in the direction `(1,0)` and '0's
          represent steps in the direction `(0,1)`. The resulting
          path will remain weakly above the diagonal `y = x`.
  
          The area statistic is the number of complete
          squares in the integer lattice which are below the path and above
          the line `y = x`. The 'half-squares' directly above the
          line `y = x` do not contribute to this statistic.
  
          EXAMPLES::
  
              sage: dw = DyckWord([1,0,1,0])
              sage: dw.area() # 2 half-squares, 0 complete squares
              0
  
          ::
  
              sage: dw = DyckWord([1,1,1,0,1,1,1,0,0,0,1,1,0,0,1,0,0,0])
              sage: dw.area()
              19
  
          ::
  
              sage: DyckWord([1,1,1,1,0,0,0,0]).area()
              6
              sage: DyckWord([1,1,1,0,1,0,0,0]).area()
              5
              sage: DyckWord([1,1,1,0,0,1,0,0]).area()
              4
              sage: DyckWord([1,1,1,0,0,0,1,0]).area()
              3
              sage: DyckWord([1,0,1,1,0,1,0,0]).area()
              2
              sage: DyckWord([1,1,0,1,1,0,0,0]).area()
              4
              sage: DyckWord([1,1,0,0,1,1,0,0]).area()
              2
              sage: DyckWord([1,0,1,1,1,0,0,0]).area()
              3
              sage: DyckWord([1,0,1,1,0,0,1,0]).area()
              1
              sage: DyckWord([1,0,1,0,1,1,0,0]).area()
              1
              sage: DyckWord([1,1,0,0,1,0,1,0]).area()
              1
              sage: DyckWord([1,1,0,1,0,0,1,0]).area()
              2
              sage: DyckWord([1,1,0,1,0,1,0,0]).area()
              3
              sage: DyckWord([1,0,1,0,1,0,1,0]).area()
              0
          """
          above = 0
          diagonal = 0
          a = 0
          for move in self:
              if move == open_symbol:
                  above += 1
              elif move == close_symbol:
                  diagonal += 1
                  a += above - diagonal
          return a
  
      def bounce_path(self):
          r"""
          Return the bounce path of ``self`` formed by starting at `(n,n)` and
          traveling West until encountering the first vertical step of ``self``,
          then South until encountering the diagonal, then West again to hit
          the path, etc. until the `(0,0)` point is reached.  The path followed
          by this walk is the bounce path.
  
          .. SEEALSO:: :meth:`bounce`
  
          EXAMPLES::
  
              sage: DyckWord([1,1,0,1,0,0]).bounce_path()
              [1, 0, 1, 1, 0, 0]
              sage: DyckWord([1,1,1,0,0,0]).bounce_path()
              [1, 1, 1, 0, 0, 0]
              sage: DyckWord([1,0,1,0,1,0]).bounce_path()
              [1, 0, 1, 0, 1, 0]
              sage: DyckWord([1,1,1,1,0,0,1,0,0,0]).bounce_path()
              [1, 1, 0, 0, 1, 1, 1, 0, 0, 0]
  
          TESTS::
  
              sage: DyckWord([]).bounce_path()
              []
              sage: DyckWord([1,0]).bounce_path()
              [1, 0]
  
          """
          area_seq = self.to_area_sequence()
          i = len(area_seq) - 1
          n = 5
          while i > 0:
              n -= 1
              a = area_seq[i]
              i_new = i - a
              while i > i_new:
                  i -= 1
                  area_seq[i] = area_seq[i + 1] - 1
              i -= 1
          return DyckWord(area_sequence=area_seq)
  
      def bounce(self):
          r"""
          Return the bounce statistic of ``self`` due to J. Haglund,
          see [Hag2008]_.
  
          One can view a balanced Dyck word as a lattice path from `(0,0)` to
          `(n,n)` in the first quadrant by letting '1's represent steps in
          the direction `(0,1)` and '0's represent steps in the direction
          `(1,0)`.  The resulting path will remain weakly above the diagonal
          `y = x`.
  
          We describe the bounce statistic of such a path in terms of what is
          known as the "bounce path".
  
          We can think of our bounce path as describing the trail of a billiard
          ball shot West from `(n, n)`, which "bounces" down whenever it
          encounters a vertical step and "bounces" left when it encounters the
          line `y = x`.
  
          The bouncing ball will strike the diagonal at the places
  
          .. MATH::
  
              (0, 0), (j_1, j_1), (j_2, j_2), \ldots, (j_r-1, j_r-1), (j_r, j_r)
              = (n, n).
  
          We define the bounce to be the sum `\sum_{i=1}^{r-1} j_i`.
  
          EXAMPLES::
  
              sage: DyckWord([1,1,1,0,1,1,1,0,0,0,1,1,0,0,1,0,0,0]).bounce()
              7
              sage: DyckWord([1,1,1,1,0,0,0,0]).bounce()
              0
              sage: DyckWord([1,1,1,0,1,0,0,0]).bounce()
              1
              sage: DyckWord([1,1,1,0,0,1,0,0]).bounce()
              2
              sage: DyckWord([1,1,1,0,0,0,1,0]).bounce()
              3
              sage: DyckWord([1,0,1,1,0,1,0,0]).bounce()
              3
              sage: DyckWord([1,1,0,1,1,0,0,0]).bounce()
              1
              sage: DyckWord([1,1,0,0,1,1,0,0]).bounce()
              2
              sage: DyckWord([1,0,1,1,1,0,0,0]).bounce()
              1
              sage: DyckWord([1,0,1,1,0,0,1,0]).bounce()
              4
              sage: DyckWord([1,0,1,0,1,1,0,0]).bounce()
              3
              sage: DyckWord([1,1,0,0,1,0,1,0]).bounce()
              5
              sage: DyckWord([1,1,0,1,0,0,1,0]).bounce()
              4
              sage: DyckWord([1,1,0,1,0,1,0,0]).bounce()
              2
              sage: DyckWord([1,0,1,0,1,0,1,0]).bounce()
              6
          """
          x_pos = len(self) // 2
          y_pos = len(self) // 2
  
          b = 0
  
          mode = "left"
          makeup_steps = 0
          l = self._list[:]
          l.reverse()
  
          for move in l:
              #print x_pos, y_pos, mode, move
              if mode == "left":
                  if move == close_symbol:
                      x_pos -= 1
                  elif move == open_symbol:
                      y_pos -= 1
                      if x_pos == y_pos:
                          b += x_pos
                      else:
                          mode = "drop"
              elif mode == "drop":
                  if move == close_symbol:
                      makeup_steps += 1
                  elif move == open_symbol:
                      y_pos -= 1
                      if x_pos == y_pos:
                          b += x_pos
                          mode = "left"
                          x_pos -= makeup_steps
                          makeup_steps = 0
  
          return b
  
      def dinv(self, labeling=None):
          r"""
          Return the dinv statistic of ``self`` due to M. Haiman, see [Hag2008]_.
  
          If a labeling is provided then this function returns the dinv of the
          labeled Dyck word.
  
          INPUT:
  
          - ``labeling`` -- an optional argument to be viewed as the labelings
            of the vertical edges of the Dyck path
  
          OUTPUT:
  
          - an integer representing the ``dinv`` statistic of the Dyck path
            or the labelled Dyck path.
  
          EXAMPLES::
  
              sage: DyckWord([1,0,1,0,1,0,1,0,1,0]).dinv()
              10
              sage: DyckWord([1,1,1,1,1,0,0,0,0,0]).dinv()
              0
              sage: DyckWord([1,1,1,1,0,1,0,0,0,0]).dinv()
              1
              sage: DyckWord([1,1,0,1,0,0,1,1,0,1,0,1,0,0]).dinv()
              13
              sage: DyckWord([1,1,0,1,0,0,1,1,0,1,0,1,0,0]).dinv([1,2,3,4,5,6,7])
              11
              sage: DyckWord([1,1,0,1,0,0,1,1,0,1,0,1,0,0]).dinv([6,7,5,3,4,2,1])
              2
          """
          alist = self.to_area_sequence()
          cnt = 0
          for j in range(len(alist)):
              for i in range(j):
                  if (alist[i] - alist[j] == 0 and (labeling is None or labeling[i] < labeling[j])) or (alist[i] - alist[j] == 1 and (labeling is None or labeling[i] > labeling[j])):
                      cnt += 1
          return cnt
  
      @combinatorial_map(name='to alternating sign matrix')
      def to_alternating_sign_matrix(self):
          r"""
          Return ``self`` as an alternating sign matrix.
  
          This is an inclusion map from Dyck words of length `2n` to certain
          `n \times n` alternating sign matrices.
  
          EXAMPLES::
  
              sage: DyckWord([1,1,1,0,1,0,0,0]).to_alternating_sign_matrix()
              [ 0  0  1  0]
              [ 1  0 -1  1]
              [ 0  1  0  0]
              [ 0  0  1  0]
              sage: DyckWord([1,0,1,0,1,1,0,0]).to_alternating_sign_matrix()
              [1 0 0 0]
              [0 1 0 0]
              [0 0 0 1]
              [0 0 1 0]
          """
          parkfn = self.reverse().to_non_decreasing_parking_function()
          parkfn2 = [len(parkfn) + 1 - parkfn[i] for i in range(len(parkfn))]
          monotone_triangle = [[0] * (len(parkfn2) - j)
                               for j in range(len(parkfn2))]
          for i in range(len(monotone_triangle)):
              for j in range(len(monotone_triangle[i])):
                  monotone_triangle[i][j] = len(monotone_triangle[i]) - j
              monotone_triangle[i][0] = parkfn2[i]
          A = AlternatingSignMatrices(len(parkfn))
          return A.from_monotone_triangle(monotone_triangle)
  
  
  class DyckWords(UniqueRepresentation, Parent):
      r"""
      Dyck words.
  
      A Dyck word is a sequence `(w_1, \ldots, w_n)` consisting of 1 s and 0 s,
      with the property that for any `i` with `1 \leq i \leq n`, the sequence
      `(w_1, \ldots, w_i)` contains at least as many 1 s as 0 s.
  
      A Dyck word is balanced if the total number of 1 s is equal to the total
      number of 0 s. The number of balanced Dyck words of length `2k` is given
      by the :func:`Catalan number<sage.combinat.combinat.catalan_number>` `C_k`.
  
      EXAMPLES:
  
      This class can be called with three keyword parameters ``k1``, ``k2``
      and ``complete``.
  
      If neither ``k1`` nor ``k2`` are specified, then :class:`DyckWords`
      returns the combinatorial class of all balanced (=complete) Dyck words,
      unless the keyword ``complete`` is set to False (in which case it
      returns the class of all Dyck words).
  
      ::
  
          sage: DW = DyckWords(); DW
          Complete Dyck words
          sage: [] in DW
          True
          sage: [1, 0, 1, 0] in DW
          True
          sage: [1, 1, 0] in DW
          False
          sage: ADW = DyckWords(complete=False); ADW
          Dyck words
          sage: [] in ADW
          True
          sage: [1, 0, 1, 0] in ADW
          True
          sage: [1, 1, 0] in ADW
          True
          sage: [1, 0, 0] in ADW
          False
  
      If just ``k1`` is specified, then it returns the balanced Dyck words with
      ``k1`` opening parentheses and ``k1`` closing parentheses.
  
      ::
  
          sage: DW2 = DyckWords(2); DW2
          Dyck words with 2 opening parentheses and 2 closing parentheses
          sage: DW2.first()
          [1, 0, 1, 0]
          sage: DW2.last()
          [1, 1, 0, 0]
          sage: DW2.cardinality()
          2
          sage: DyckWords(100).cardinality() == catalan_number(100)
          True
  
      If ``k2`` is specified in addition to ``k1``, then it returns the
      Dyck words with ``k1`` opening parentheses and ``k2`` closing parentheses.
  
      ::
  
          sage: DW32 = DyckWords(3,2); DW32
          Dyck words with 3 opening parentheses and 2 closing parentheses
          sage: DW32.list()
          [[1, 0, 1, 0, 1],
           [1, 0, 1, 1, 0],
           [1, 1, 0, 0, 1],
           [1, 1, 0, 1, 0],
           [1, 1, 1, 0, 0]]
      """
      @staticmethod
      def __classcall_private__(cls, k1=None, k2=None, complete=True):
          """
          Choose the correct parent based upon input.
  
          EXAMPLES::
  
              sage: DW1 = DyckWords(3,3)
              sage: DW2 = DyckWords(3)
              sage: DW1 is DW2
              True
          """
          if k2 is None:
              if k1 is None:
                  if complete:
                      return CompleteDyckWords_all()
                  return DyckWords_all()
  
              k1 = ZZ(k1)
              if k1 < 0:
                  raise ValueError("k1 (= %s) must be nonnegative" % k1)
              return CompleteDyckWords_size(k1)
          else:
              k1 = ZZ(k1)
  
          k2 = ZZ(k2)
          if k1 < 0 or (k2 is not None and k2 < 0):
              raise ValueError("k1 (= %s) and k2 (= %s) must be nonnegative, with k1 >= k2." % (k1, k2))
          if k1 < k2:
              raise ValueError("k1 (= %s) must be >= k2 (= %s)" % (k1, k2))
  
          if k1 == k2:
              return CompleteDyckWords_size(k1)
          return DyckWords_size(k1, k2)
  
      Element = DyckWord
      global_options = DyckWordOptions
  
      def _element_constructor_(self, word):
          """
          Construct an element of ``self``.
  
          EXAMPLES::
  
              sage: D = DyckWords()
              sage: elt = D([1, 1, 0, 1, 0, 0]); elt
              [1, 1, 0, 1, 0, 0]
              sage: elt.parent() is D
              True
          """
          if isinstance(word, DyckWord) and word.parent() is self:
              return word
          return self.element_class(self, list(word))
  
      def __contains__(self, x):
          r"""
          TESTS::
  
              sage: D = DyckWords(complete=False)
              sage: [] in D
              True
              sage: [1] in D
              True
              sage: [0] in D
              False
              sage: [1, 0] in D
              True
          """
          if isinstance(x, DyckWord):
              return True
  
          if not isinstance(x, list):
              return False
  
          return is_a(x)
  
      def from_heights(self, heights):
          r"""
          Compute a Dyck word knowing its heights.
  
          We view the Dyck word as a Dyck path from `(0, 0)` to
          `(2n, 0)` in the first quadrant by letting ``1``'s represent
          steps in the direction `(1, 1)` and ``0``'s represent steps in
          the direction `(1, -1)`.
  
          The :meth:`heights` is the sequence of the `y`-coordinates of
          the `2n+1` lattice points along this path.
  
          EXAMPLES::
  
              sage: from sage.combinat.dyck_word import DyckWord
              sage: D = DyckWords(complete=False)
              sage: D.from_heights((0,))
              []
              sage: D.from_heights((0, 1, 0))
              [1, 0]
              sage: D.from_heights((0, 1, 2, 1, 0))
              [1, 1, 0, 0]
  
          This also works for incomplete Dyck words::
  
              sage: D.from_heights((0, 1, 2, 1, 2, 1))
              [1, 1, 0, 1, 0]
              sage: D.from_heights((0, 1, 2, 1))
              [1, 1, 0]
  
          .. SEEALSO:: :meth:`heights`, :meth:`min_from_heights`
  
          TESTS::
  
              sage: all(dw == D.from_heights(dw.heights())
              ....:     for i in range(7) for dw in DyckWords(i))
              True
  
              sage: D.from_heights((1, 2, 1))
              Traceback (most recent call last):
              ...
              ValueError: heights must start with 0: (1, 2, 1)
              sage: D.from_heights((0, 1, 4, 1))
              Traceback (most recent call last):
              ...
              ValueError: consecutive heights must differ by exactly 1: (0, 1, 4, 1)
              sage: D.from_heights(())
              Traceback (most recent call last):
              ...
              ValueError: heights must start with 0: ()
          """
          l1 = len(heights)-1
          res = [0]*(l1)
          if heights == () or heights[0] != 0:
              raise ValueError("heights must start with 0: %s" % (heights,))
          for i in range(l1):
              if heights[i] == heights[i+1]-1:
                  res[i] = 1
              elif heights[i] != heights[i+1]+1:
                  raise ValueError("consecutive heights must differ by exactly 1: %s" % (heights,))
          return self.element_class(self, res)
  
      def min_from_heights(self, heights):
          r"""
          Compute the smallest Dyck word which achieves or surpasses
          a given sequence of heights.
  
          INPUT:
  
          - ``heights`` -- a nonempty list or iterable consisting of
            nonnegative integers, the first of which is `0`
  
          OUTPUT:
  
          - The smallest Dyck word whose sequence of heights is
            componentwise higher-or-equal to ``heights``. Here,
            "smaller" can be understood both in the sense of
            lexicographic order on the Dyck words, and in the sense
            of every vertex of the path having the smallest possible
            height.
  
          .. SEEALSO::
  
              - :meth:`heights`
              - :meth:`from_heights`
  
          EXAMPLES::
  
              sage: D = DyckWords(complete=False)
              sage: D.min_from_heights((0,))
              []
              sage: D.min_from_heights((0, 1, 0))
              [1, 0]
              sage: D.min_from_heights((0, 0, 2, 0, 0))
              [1, 1, 0, 0]
              sage: D.min_from_heights((0, 0, 2, 0, 2, 0))
              [1, 1, 0, 1, 0]
              sage: D.min_from_heights((0, 0, 1, 0, 1, 0))
              [1, 1, 0, 1, 0]
  
          TESTS::
  
              sage: D.min_from_heights(())
              Traceback (most recent call last):
              ...
              ValueError: heights must start with 0: ()
          """
          if heights == () or heights[0] != 0:
              raise ValueError("heights must start with 0: %s" % (heights,))
          # round heights to the smallest even-odd integer
          heights = list(heights)
          for i in range(0, len(heights), 2):
              if heights[i] % 2 == 1:
                  heights[i] += 1
          for i in range(1, len(heights), 2):
              if heights[i] % 2 == 0:
                  heights[i] += 1
  
          # smooth heights
          for i in range(len(heights) - 1):
              if heights[i + 1] < heights[i]:
                  heights[i + 1] = heights[i] - 1
          for i in range(len(heights)-1, 0, -1):
              if heights[i] > heights[i - 1]:
                  heights[i-1] = heights[i] - 1
          return self.from_heights(heights)
  
  
  class DyckWords_all(DyckWords):
      """
      All Dyck words.
      """
      def __init__(self):
          """
          Intialize ``self``.
  
          EXAMPLES::
  
              sage: TestSuite(DyckWords(complete=False)).run()
          """
          DyckWords.__init__(self, category=InfiniteEnumeratedSets())
  
      def _repr_(self):
          r"""
          TESTS::
  
              sage: DyckWords(complete=False)
              Dyck words
          """
          return "Dyck words"
  
      def _an_element_(self):
          r"""
          TESTS::
  
              sage: DyckWords(complete=False).an_element()
              [1, 0, 1, 0, 1, 0, 1, 0, 1, 0]
          """
          return DyckWords(5).an_element()
  
      def __iter__(self):
          """
          Iterate over ``self``.
  
          EXAMPLES::
  
              sage: it = DyckWords(complete=False).__iter__()
              sage: [next(it) for x in range(10)]
              [[],
               [1],
               [1, 0],
               [1, 1],
               [1, 0, 0],
               [1, 0, 1],
               [1, 1, 0],
               [1, 1, 1],
               [1, 0, 1, 0],
               [1, 1, 0, 0]]
          """
          n = 0
          yield self.element_class(self, [])
          while True:
              for k in range(1, n+1):
                  for x in DyckWords_size(k, n-k):
                      yield self.element_class(self, list(x))
              n += 1
  
  
  class DyckWordBacktracker(GenericBacktracker):
      r"""
      This class is an iterator for all Dyck words
      with `n` opening parentheses and `n - k` closing parentheses using
      the backtracker class. It is used by the :class:`DyckWords_size` class.
  
      This is not really meant to be called directly, partially because
      it fails in a couple corner cases: ``DWB(0)`` yields ``[0]``, not the
      empty word, and ``DWB(k, k+1)`` yields something (it shouldn't yield
      anything). This could be fixed with a sanity check in ``_rec()``, but
      then we'd be doing the sanity check *every time* we generate new
      objects; instead, we do *one* sanity check in :class:`DyckWords` and
      assume here that the sanity check has already been made.
  
      AUTHOR:
  
      - Dan Drake (2008-05-30)
      """
      def __init__(self, k1, k2):
          r"""
          TESTS::
  
              sage: from sage.combinat.dyck_word import DyckWordBacktracker
              sage: len(list(DyckWordBacktracker(5, 5)))
              42
              sage: len(list(DyckWordBacktracker(6,4)))
              90
              sage: len(list(DyckWordBacktracker(7,0)))
              1
          """
          GenericBacktracker.__init__(self, [], (0, 0))
          # note that the comments in this class think of our objects as
          # Dyck paths, not words; having k1 opening parens and k2 closing
          # parens corresponds to paths of length k1 + k2 ending at height
          # k1 - k2.
          k1 = ZZ(k1)
          k2 = ZZ(k2)
          self.n = k1 + k2
          self.endht = k1 - k2
  
      def _rec(self, path, state):
          r"""
          TESTS::
  
              sage: from sage.combinat.dyck_word import DyckWordBacktracker
              sage: dwb = DyckWordBacktracker(3, 3)
              sage: list(dwb._rec([1,1,0],(3, 2)))
              [([1, 1, 0, 0], (4, 1), False), ([1, 1, 0, 1], (4, 3), False)]
              sage: list(dwb._rec([1,1,0,0],(4, 0)))
              [([1, 1, 0, 0, 1], (5, 1), False)]
              sage: list(DyckWordBacktracker(4, 4)._rec([1,1,1,1],(4, 4)))
              [([1, 1, 1, 1, 0], (5, 3), False)]
          """
          len, ht = state
  
          if len < self.n - 1:
              # if length is less than n-1, new path won't have length n, so
              # don't yield it, and keep building paths
  
              # if the path isn't too low and is not touching the x-axis, we can
              # yield a path with a downstep at the end
              if ht > (self.endht - (self.n - len)) and ht > 0:
                  yield path + [0], (len + 1, ht - 1), False
  
              # if the path isn't too high, it can also take an upstep
              if ht < (self.endht + (self.n - len)):
                  yield path + [1], (len + 1, ht + 1), False
          else:
              # length is n - 1, so add a single step (up or down,
              # according to current height and endht), don't try to
              # construct more paths, and yield the path
              if ht < self.endht:
                  yield path + [1], None, True
              else:
                  yield path + [0], None, True
  
  
  class DyckWords_size(DyckWords):
      """
      Dyck words with `k_1` openers and `k_2` closers.
      """
      def __init__(self, k1, k2):
          r"""
          TESTS:
  
          Check that :trac:`18244` is fixed::
  
              sage: DyckWords(13r, 8r).cardinality()
              87210
              sage: parent(_)
              Integer Ring
              sage: TestSuite(DyckWords(4,2)).run()
          """
          self.k1 = ZZ(k1)
          self.k2 = ZZ(k2)
          DyckWords.__init__(self, category=FiniteEnumeratedSets())
  
      def _repr_(self):
          r"""
          TESTS::
  
              sage: DyckWords(4)
              Dyck words with 4 opening parentheses and 4 closing parentheses
          """
          return "Dyck words with %s opening parentheses and %s closing parentheses" % (self.k1, self.k2)
  
      def __contains__(self, x):
          r"""
          EXAMPLES::
  
              sage: [1, 0, 0, 1] in DyckWords(2,2)
              False
              sage: [1, 0, 1, 0] in DyckWords(2,2)
              True
              sage: [1, 0, 1, 0, 1] in DyckWords(3,2)
              True
              sage: [1, 0, 1, 1, 0] in DyckWords(3,2)
              True
              sage: [1, 0, 1, 1] in DyckWords(3,1)
              True
          """
          return is_a(x, self.k1, self.k2)
  
      def __iter__(self):
          r"""
          Return an iterator for Dyck words with ``k1`` opening and ``k2``
          closing parentheses.
  
          EXAMPLES::
  
              sage: list(DyckWords(0))
              [[]]
              sage: list(DyckWords(1))
              [[1, 0]]
              sage: list(DyckWords(2))
              [[1, 0, 1, 0], [1, 1, 0, 0]]
              sage: len(DyckWords(5))
              42
              sage: list(DyckWords(3,2))
              [[1, 0, 1, 0, 1],
               [1, 0, 1, 1, 0],
               [1, 1, 0, 0, 1],
               [1, 1, 0, 1, 0],
               [1, 1, 1, 0, 0]]
          """
          if self.k1 == 0:
              yield self.element_class(self, [])
          elif self.k2 == 0:
              yield self.element_class(self, [open_symbol]*self.k1)
          else:
              for w in DyckWordBacktracker(self.k1, self.k2):
                  yield self.element_class(self, w)
  
      def cardinality(self):
          r"""
          Return the number of Dyck words with `k_1` openers and `k_2` closers.
  
          This number is
  
          .. MATH::
  
              \frac{k_1 - k_2 + 1}{k_1 + 1} \binom{k_1 + k_2}{k_2}
              = \binom{k_1 + k_2}{k_2} - \binom{k_1 + k_2}{k_2 - 1}
  
          if `k_2 \leq k_1 + 1`, and `0` if `k_2 > k_1` (these numbers are the
          same if `k_2 = k_1 + 1`).
  
          EXAMPLES::
  
              sage: DyckWords(3, 2).cardinality()
              5
              sage: all(all(DyckWords(p, q).cardinality()
              ....:           == len(DyckWords(p, q).list()) for q in range(p + 1))
              ....:      for p in range(7))
              True
          """
          from sage.arith.all import binomial
          return (self.k1 - self.k2 + 1) * binomial(self.k1 + self.k2, self.k2) // (self.k1 + 1)
  
  ################################################################
  ## Complete Dyck words
  
  
  class CompleteDyckWords(DyckWords):
      """
      Abstract base class for all complete Dyck words.
      """
      Element = DyckWord_complete
  
      def __contains__(self, x):
          r"""
          TESTS::
  
              sage: [] in DyckWords()
              True
              sage: [1] in DyckWords()
              False
              sage: [0] in DyckWords()
              False
          """
          if isinstance(x, DyckWord_complete):
              return True
  
          if not isinstance(x, list):
              return False
  
          if len(x) % 2 != 0:
              return False
  
          return is_a(x, len(x) // 2)
  
      def from_Catalan_code(self, code):
          r"""
          Return the Dyck word associated to the given Catalan code
          ``code``.
  
          A Catalan code of length `n` is a sequence
          `(a_1, a_2, \ldots, a_n)` of `n` integers `a_i` such that:
  
          - `0 \leq a_i \leq n-i` for every `i`;
  
          - if `i < j` and `a_i > 0` and `a_j > 0` and
            `a_{i+1} = a_{i+2} = \cdots = a_{j-1} = 0`,
            then `a_i - a_j < j-i`.
  
          It turns out that the Catalan codes of length `n` are in
          bijection with Dyck words.
  
          The Catalan code of a Dyck word is example (x) in Richard Stanley's
          exercises on combinatorial interpretations for Catalan objects.
          The code in this example is the reverse of the description provided
          there. See [Sta-EC2]_ and [StaCat98]_.
  
          EXAMPLES::
  
              sage: DyckWords().from_Catalan_code([])
              []
              sage: DyckWords().from_Catalan_code([0])
              [1, 0]
              sage: DyckWords().from_Catalan_code([0, 1])
              [1, 1, 0, 0]
              sage: DyckWords().from_Catalan_code([0, 0])
              [1, 0, 1, 0]
          """
          code = list(code)
          if not code:
              return self.element_class(self, [])
          res = self.from_Catalan_code(code[:-1])
          cuts = [0] + res.returns_to_zero()
          lst = [1] + res[:cuts[code[-1]]] + [0] + res[cuts[code[-1]]:]
          return self.element_class(self, lst)
  
      def from_area_sequence(self, code):
          r"""
          Return the Dyck word associated to the given area sequence
          ``code``.
  
          See :meth:`to_area_sequence` for a definition of the area
          sequence of a Dyck word.
  
          .. SEEALSO:: :meth:`area`, :meth:`to_area_sequence`.
  
          INPUT:
  
          - ``code`` -- a list of integers satisfying ``code[0] == 0``
            and ``0 <= code[i+1] <= code[i]+1``.
  
          EXAMPLES::
  
              sage: DyckWords().from_area_sequence([])
              []
              sage: DyckWords().from_area_sequence([0])
              [1, 0]
              sage: DyckWords().from_area_sequence([0, 1])
              [1, 1, 0, 0]
              sage: DyckWords().from_area_sequence([0, 0])
              [1, 0, 1, 0]
          """
          if not is_area_sequence(code):
              raise ValueError("The given sequence is not a sequence giving "
                               "the number of cells between the Dyck path "
                               "and the diagonal.")
          dyck_word = []
          for i in xrange(len(code)):
              if i > 0:
                  dyck_word.extend([close_symbol]*(code[i-1]-code[i]+1))
              dyck_word.append(open_symbol)
          dyck_word.extend([close_symbol]*(2*len(code)-len(dyck_word)))
          return self.element_class(self, dyck_word)
  
      def from_noncrossing_partition(self, ncp):
          r"""
          Convert a noncrossing partition ``ncp`` to a Dyck word.
  
          TESTS::
  
              sage: DyckWord(noncrossing_partition=[[1,2]]) # indirect doctest
              [1, 1, 0, 0]
              sage: DyckWord(noncrossing_partition=[[1],[2]])
              [1, 0, 1, 0]
  
          ::
  
              sage: dws = DyckWords(5).list()
              sage: ncps = map(lambda x: x.to_noncrossing_partition(), dws)
              sage: dws2 = map(lambda x: DyckWord(noncrossing_partition=x), ncps)
              sage: dws == dws2
              True
          """
          l = [0] * int(sum([len(v) for v in ncp]))
          for v in ncp:
              l[v[-1] - 1] = len(v)
  
          res = []
          for i in l:
              res += [open_symbol] + [close_symbol]*int(i)
          return self.element_class(self, res)
  
      def from_non_decreasing_parking_function(self, pf):
          r"""
          Bijection from :class:`non-decreasing parking
          functions<sage.combinat.non_decreasing_parking_function.NonDecreasingParkingFunctions>`.
          See there the method
          :meth:`~sage.combinat.non_decreasing_parking_function.NonDecreasingParkingFunction.to_dyck_word`
          for more information.
  
          EXAMPLES::
  
              sage: D = DyckWords()
              sage: D.from_non_decreasing_parking_function([])
              []
              sage: D.from_non_decreasing_parking_function([1])
              [1, 0]
              sage: D.from_non_decreasing_parking_function([1,1])
              [1, 1, 0, 0]
              sage: D.from_non_decreasing_parking_function([1,2])
              [1, 0, 1, 0]
              sage: D.from_non_decreasing_parking_function([1,1,1])
              [1, 1, 1, 0, 0, 0]
              sage: D.from_non_decreasing_parking_function([1,2,3])
              [1, 0, 1, 0, 1, 0]
              sage: D.from_non_decreasing_parking_function([1,1,3,3,4,6,6])
              [1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0]
  
          TESTS::
  
              sage: D.from_non_decreasing_parking_function(NonDecreasingParkingFunction([]))
              []
              sage: D.from_non_decreasing_parking_function(NonDecreasingParkingFunction([1,1,3,3,4,6,6]))
              [1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0]
          """
          return self.from_area_sequence([i - pf[i] + 1 for i in range(len(pf))])
  
  
  class CompleteDyckWords_all(CompleteDyckWords, DyckWords_all):
      """
      All complete Dyck words.
      """
      def _repr_(self):
          r"""
          TESTS::
  
              sage: DyckWords()
              Complete Dyck words
          """
          return "Complete Dyck words"
  
      def __iter__(self):
          """
          Iterate over ``self``.
  
          EXAMPLES::
  
              sage: it = DyckWords().__iter__()
              sage: [next(it) for x in range(10)]
              [[],
               [1, 0],
               [1, 0, 1, 0],
               [1, 1, 0, 0],
               [1, 0, 1, 0, 1, 0],
               [1, 0, 1, 1, 0, 0],
               [1, 1, 0, 0, 1, 0],
               [1, 1, 0, 1, 0, 0],
               [1, 1, 1, 0, 0, 0],
               [1, 0, 1, 0, 1, 0, 1, 0]]
          """
          n = 0
          while True:
              for x in CompleteDyckWords_size(n):
                  yield self.element_class(self, list(x))
              n += 1
  
      class height_poset(UniqueRepresentation, Parent):
          r"""
          The poset of complete Dyck words compared componentwise by
          ``heights``.
          This is, ``D`` is smaller than or equal to ``D'`` if it is
          weakly below ``D'``.
          """
          def __init__(self):
              r"""
              TESTS::
  
                  sage: poset = DyckWords().height_poset()
                  sage: TestSuite(poset).run()
              """
              Parent.__init__(self, facade=DyckWords_all(), category=Posets())
  
          def _an_element_(self):
              r"""
              TESTS::
  
                  sage: DyckWords().height_poset().an_element()   # indirect doctest
                  [1, 0, 1, 0, 1, 0, 1, 0, 1, 0]
              """
              return DyckWords_all().an_element()
  
          def __call__(self, obj):
              r"""
              TESTS::
  
                  sage: poset = DyckWords().height_poset()
                  sage: poset([1,0,1,0])
                  [1, 0, 1, 0]
              """
              return DyckWords_all()(obj)
  
          def le(self, dw1, dw2):
              r"""
              Compare two Dyck words of equal size, and return ``True`` if
              all of the heights of ``dw1`` are less than or equal to the
              respective heights of ``dw2`` .
  
              .. SEEALSO::
  
                  :meth:`heights<sage.combinat.dyck_word.DyckWord.heights>`
  
              EXAMPLES::
  
                  sage: poset = DyckWords().height_poset()
                  sage: poset.le(DyckWord([]), DyckWord([]))
                  True
                  sage: poset.le(DyckWord([1,0]), DyckWord([1,0]))
                  True
                  sage: poset.le(DyckWord([1,0,1,0]), DyckWord([1,1,0,0]))
                  True
                  sage: poset.le(DyckWord([1,1,0,0]), DyckWord([1,0,1,0]))
                  False
                  sage: [poset.le(dw1, dw2)
                  ....:     for dw1 in DyckWords(3) for dw2 in DyckWords(3)]
                  [True, True, True, True, True, False, True, False, True, True,
                  False, False, True, True, True, False, False, False, True,
                  True, False, False, False, False, True]
              """
              if len(dw1) != len(dw2):
                  raise ValueError("Length mismatch: %s and %s" % (dw1, dw2))
              sh = dw1.heights()
              oh = dw2.heights()
              return all(sh[i] <= oh[i] for i in range(len(dw1)))
  
  
  class CompleteDyckWords_size(CompleteDyckWords, DyckWords_size):
      """
      All complete Dyck words of a given size.
      """
      def __init__(self, k):
          """
          Initialize ``self``.
  
          TESTS::
  
              sage: TestSuite(DyckWords(4)).run()
          """
          CompleteDyckWords.__init__(self, category=FiniteEnumeratedSets())
          DyckWords_size.__init__(self, k, k)
  
      def __contains__(self, x):
          r"""
          TESTS::
  
              sage: [1, 0] in DyckWords(1)
              True
              sage: [1, 0] in DyckWords(2)
              False
              sage: [1, 1, 0, 0] in DyckWords(2)
              True
              sage: [1, 0, 0, 1] in DyckWords(2)
              False
          """
          return CompleteDyckWords.__contains__(self, x) and len(x) // 2 == self.k1
  
      def cardinality(self):
          r"""
          Return the number of complete Dyck words of semilength `n`, i.e. the
          `n`-th :func:`Catalan number<sage.combinat.combinat.catalan_number>`.
  
          EXAMPLES::
  
              sage: DyckWords(4).cardinality()
              14
              sage: ns = range(9)
              sage: dws = [DyckWords(n) for n in ns]
              sage: all([dw.cardinality() == len(dw.list()) for dw in dws])
              True
          """
          return catalan_number(self.k1)
  
      def random_element(self):
          """
          Return a random complete Dyck word of semilength `n`
  
          The algorithm is based on a classical combinatorial fact. One
          chooses at random a word with `n` 0's and `n+1` 1's. One then
          considers every 1 as an ascending step and every 0 as a
          descending step, and one finds the lowest point of the path
          (with respect to a slightly tilted slope). One then cuts the
          path at this point and builds a Dyck word by exchanging the
          two parts of the word and removing the initial step.
  
          .. TODO::
  
              extend this to m-Dyck words
  
          EXAMPLES::
  
              sage: dw = DyckWords(8)
              sage: dw.random_element()  # random
              [1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0]
  
              sage: D = DyckWords(8)
              sage: D.random_element() in D
              True
          """
          from sage.misc.prandom import shuffle
          n = self.k1
          w = [0] * n + [1] * (n + 1)
          shuffle(w)
          idx = 0
          height = 0
          height_min = 0
          for i in range(2 * n):
              if w[i] == 1:
                  height += n
              else:
                  height -= n + 1
                  if height < height_min:
                      height_min = height
                      idx = i + 1
          w = w[idx:] + w[:idx]
          return self.element_class(self, w[1:])
  
      def _iter_by_recursion(self):
          """
          Iterate over ``self`` by recursively using the position of
          the first return to 0.
  
          EXAMPLES::
  
              sage: DW = DyckWords(4)
              sage: L = [d for d in DW._iter_by_recursion()]; L
              [[1, 0, 1, 0, 1, 0, 1, 0],
               [1, 0, 1, 0, 1, 1, 0, 0],
               [1, 0, 1, 1, 0, 0, 1, 0],
               [1, 0, 1, 1, 0, 1, 0, 0],
               [1, 0, 1, 1, 1, 0, 0, 0],
               [1, 1, 0, 0, 1, 0, 1, 0],
               [1, 1, 0, 0, 1, 1, 0, 0],
               [1, 1, 0, 1, 0, 0, 1, 0],
               [1, 1, 1, 0, 0, 0, 1, 0],
               [1, 1, 0, 1, 0, 1, 0, 0],
               [1, 1, 0, 1, 1, 0, 0, 0],
               [1, 1, 1, 0, 0, 1, 0, 0],
               [1, 1, 1, 0, 1, 0, 0, 0],
               [1, 1, 1, 1, 0, 0, 0, 0]]
              sage: len(L) == DW.cardinality()
              True
          """
          # Do a couple of small cases first
          if self.k1 <= 2:
              if self.k1 == 0:
                  yield self.element_class(self, [])
              elif self.k1 == 1:
                  yield self.element_class(self, [1, 0])
              elif self.k1 == 2:
                  yield self.element_class(self, [1, 0, 1, 0])
                  yield self.element_class(self, [1, 1, 0, 0])
              return
  
          # Create all necessary parents
          P = [CompleteDyckWords_size(i) for i in range(self.k1)]
  
          for i in range(self.k1):
              for p in P[i]._iter_by_recursion():
                  list_1p0 = [1] + list(p) + [0]
                  for s in P[-i-1]._iter_by_recursion():
                      yield self.element_class(self, list_1p0 + list(s))
  
  
  def is_area_sequence(seq):
      r"""
      Test if a sequence `l` of integers satisfies `l_0 = 0` and
      `0 \leq l_{i+1} \leq l_i + 1` for `i > 0`.
  
      EXAMPLES::
  
          sage: from sage.combinat.dyck_word import is_area_sequence
          sage: is_area_sequence([0,2,0])
          False
          sage: is_area_sequence([1,2,3])
          False
          sage: is_area_sequence([0,1,0])
          True
          sage: is_area_sequence([0,1,2])
          True
          sage: is_area_sequence([])
          True
      """
      if seq == []:
          return True
      return seq[0] == 0 and all(0 <= seq[i+1] and seq[i+1] <= seq[i]+1
                                 for i in xrange(len(seq)-1))
  
  
  def is_a(obj, k1=None, k2=None):
      r"""
      Test if ``obj`` is a Dyck word with exactly ``k1`` open symbols and
      exactly ``k2`` close symbols.
  
      If ``k1`` is not specified, then the number of open symbols can be
      arbitrary. If ``k1`` is specified but ``k2`` is not, then ``k2`` is
      set to be ``k1``.
  
      EXAMPLES::
  
          sage: from sage.combinat.dyck_word import is_a
          sage: is_a([1,1,0,0])
          True
          sage: is_a([1,0,1,0])
          True
          sage: is_a([1,1,0,0], 2)
          True
          sage: is_a([1,1,0,0], 3)
          False
          sage: is_a([1,1,0,0], 3, 2)
          False
          sage: is_a([1,1,0])
          True
          sage: is_a([0,1,0])
          False
          sage: is_a([1,0,0])
          False
          sage: is_a([1,1,0],2,1)
          True
          sage: is_a([1,1,0],2)
          False
          sage: is_a([1,1,0],1,1)
          False
      """
      if k1 is not None:
          if k2 is None:
              k2 = k1
          elif k1 < k2:
              raise ValueError("k1 (= %s) must be >= k2 (= %s)" % (k1, k2))
  
      n_opens = 0
      n_closes = 0
  
      for p in obj:
          if p == open_symbol:
              n_opens += 1
          elif p == close_symbol:
              if n_opens == n_closes:
                  return False
              n_closes += 1
          else:
              return False
  
      return (k1 is None and k2 is None) or (n_opens == k1 and n_closes == k2)
  
  
  def from_ordered_tree(tree):
      r"""
      TESTS::
  
          sage: sage.combinat.dyck_word.from_ordered_tree(1)
          Traceback (most recent call last):
          ...
          NotImplementedError: TODO
      """
      raise NotImplementedError("TODO")
  
  
  def pealing(D, return_touches=False):
      r"""
      A helper function for computing the bijection from a Dyck word to a
      `231`-avoiding permutation using the bijection "Stump".  For details
      see [Stu2008]_.
  
      .. SEEALSO::
  
          :meth:`~sage.combinat.dyck_word.DyckWord_complete.to_noncrossing_partition`
  
      EXAMPLES::
  
          sage: from sage.combinat.dyck_word import pealing
          sage: pealing(DyckWord([1,1,0,0]))
          [1, 0, 1, 0]
          sage: pealing(DyckWord([1,0,1,0]))
          [1, 0, 1, 0]
          sage: pealing(DyckWord([1, 1, 0, 0, 1, 1, 1, 0, 0, 0]))
          [1, 0, 1, 0, 1, 0, 1, 0, 1, 0]
          sage: pealing(DyckWord([1,1,0,0]),return_touches=True)
          ([1, 0, 1, 0], [[1, 2]])
          sage: pealing(DyckWord([1,0,1,0]),return_touches=True)
          ([1, 0, 1, 0], [])
          sage: pealing(DyckWord([1, 1, 0, 0, 1, 1, 1, 0, 0, 0]),return_touches=True)
          ([1, 0, 1, 0, 1, 0, 1, 0, 1, 0], [[1, 2], [3, 5]])
      """
      n = D.semilength()
      area = D.to_area_sequence()
      new_area = []
      touch_sequences = []
      touches = []
      for i in range(n-1):
          if area[i+1] == 0:
              touches.append(i+1)
              if len(touches) > 1:
                  touch_sequences.append(touches)
                  touches = []
              elif area[i] == 0:
                  touches = []
              new_area.append(0)
          elif area[i+1] == 1:
              new_area.append(0)
              touches.append(i+1)
          else:
              new_area.append(area[i+1]-2)
      new_area.append(0)
      if area[n - 1] != 0:
          touches.append(n)
          if len(touches) > 1:
              touch_sequences.append(touches)
      D = DyckWords().from_area_sequence(new_area)
      if return_touches:
          return (D, touch_sequences)
      else:
          return D
  
  from sage.structure.sage_object import register_unpickle_override
  register_unpickle_override('sage.combinat.dyck_word', 'DyckWord', DyckWord)
  