Vexillary Elements in the Hyperoctahedral Group.
In analogy with the symmetric group, we define the vexillary elements in the hyperoctahedral group to be those for which the Stanley function is a single Schur Q-function. We show that the vexillary elements can be again determined by pattern avoidance conditions. These results can be extended to include the root systems of types A, B, C, and D. Finally, we give an algorithm for multiplication of Schur Q -functions with a superfied Schur function and a method for determining the shape of a vexillary signed permutation using de taquin.
Introduction
The vexillary permutations in the symmetric group have interesting connections with
the number of reduced words, the Littlewood-Richardson rule, Stanley symmetric func-
tions, Schubert polynomials and the Schubert calculus. Lascoux and Sch-utzenberger [14]
have shown that vexillary permutations are characterized by the property that they avoid
any subsequence of length 4 with the same relative order as 2143. Macdonald has given
a good overview of vexillary permutations in [16]. In this paper we propose a definition
for vexillary elements in the hyperoctahedral group. We show that the vexillary elements
can again be determined by pattern avoidance conditions.
We will begin by reviewing the history of the Stanley symmetric functions and establishing
our notation. We have included several propositions from the literature that
we will use in the proof of the main theorem. In Section 2 we will define the vexillary
elements in the symmetric group and the hyperoctahedral group. We state and prove
that the vexillary elements are precisely those elements which avoid different patterns
of lengths 3 and 4. Due to the quantity of cases that need to be analyzed we have used
a computer to verify a key lemma in the proof of the main theorem. The definition of
vexillary can be extended to cover the root systems of type A, B, C, and D; in all four
cases the definition is equivalent to avoiding certain patterns. In Sections 3, we give an
algorithm for multiplication of Schur Q-functions with a superfied Schur function. In
Section 4 we outline a method for determining the shape of a signed permutation using
de taquin. We conclude with several open problems related to vexillary elements in
the hyperoctahedral group.
Let S n be the symmetric group whose elements are permutations written in one-line
notation as [w 1 generated by the adjacent transpositions oe i for
Date: June 24, 1996.
The first author is supported by the National Science Foundation and the University of California,
Presidential Postdoctoral Fellowship.
positions i and when acting on the right, i.e.,
be the hyperoctahedral group (or signed permutation group). The elements of
are permutations with a sign attached to every entry. We use the compact notation
where a bar is written over an element with a negative sign. For example
is generated by the adjacent transpositions oe i for along with oe 0
which acts on the right by changing the sign of the first element, i.e., [w
If w can be written as a product of the generators oe a 1
oe a 2
ap and p is minimal then
the concatenation of the indices a 1 a is a reduced word for w, and p is the length
of w, denoted l(w). Let R(w) be the set of all reduced words for w. The signed (or
unsigned) permutations [w have the same set
of reduced words. For our purposes it will useful to consider these signed permutations
as the same in the infinite groups
Let s - be the Schur function of shape - and let Q - be the Q-Schur function of shape
-. See [15] for definitions of these symmetric functions.
Definition 1. For w define the S n Stanley symmetric function by
where A(D(a)) is the set of all weakly increasing sequences such that if i then we
don't have a k\Gamma1 ? a k , (i.e. no descent in the corresponding reduced word).
For define the C n Stanley symmetric function by
where A(P (a)) is the set of all weakly increasing sequences such that if i
then we don't have a no peak in the corresponding reduced word).
In [20], Stanley showed that Gw is a symmetric function and used it to express the
number of reduced words of a permutation w in terms of f - the number of standard
tableaux of shape -, namely
where ff -
w is the coefficient of s - in Gw . Bijective proofs of (1.3) were given independently
by Lascoux and Sch-utzenberger [13] and Edelman and Greene [4]. Reiner and Shimozono
[18] have given a new interpretation of the coefficients ff -
w in terms of D(w)-peelable
tableaux.
VEXILLARY ELEMENTS IN THE HYPEROCTAHEDRAL GROUP 3
Stanley also conjectured that there should be an analog of (1.3) for B n . This conjecture
was proved independently by Haiman [7] and Kra'skiewicz [8] in the following form:
where g - is the number of standard tableaux on the shifted shape -, and fi -
w are the
coefficients of Q - when Fw is expanded in terms of the Schur Q-functions.
The Stanley symmetric functions can also be defined using the nilCoxeter algebra of S n
and B n respectively(see [5] and [6]). The relationship between Kra'skiewicz's proof of (1.4)
and B n Stanley symmetric functions are explored in [11]. See also [2, 10, 23] for other
connections to Stanley symmetric functions. The functions Fw are usually referred to as
the Stanley symmetric functions of type C because they are related to the root systems
of type C. The Weyl group for the root systems of type B and C are isomorphic, so we
can study the group B n by studying either root system. We extend the results of the
main theorem to the root systems of type B and D at the end of Section 2.
We will use these symmetric functions to define vexillary elements in S n and B n .
The Stanley functions Fw can easily be computed using Proposition 1.1 below which
is stated in terms of special elements in B n . There are two types of "transpositions"
in the hyperoctahedral group. These transpositions correspond with reflections in the
Weyl group of the root system B n . Let t ij be a transposition of the usual type i.e.
be a transposition of two
elements that also switches sign
element s ii simply changes the sign of the ith element. Let - ij be a transposition of
either type. A signed permutation w is said to have a descent at r if w r ? w r+1 .
Proposition 1.1. [1] The Stanley symmetric functions of type C have the following
recursive formulas:
0!i!r
l(wtrs t ir )=l(w)
Fwtrs t ir
l(wtrss ir )=l(w)
Fwtrss ir
where r is the last descent of w, and s is the largest position such that w s ! w r . The
recursion terminates when w is strictly increasing in which case is the
partition obtained from arranging fjw decreasing order.
For example, let is a descent and w
and This implies wt rs 3] and we have
Continuing to expand the right hand side we see [ - 4; - 3; 1; 2] is strictly increasing so
Note that l(wt rs ) always equal Proposition 1.1 because of the choice for r
and s. If l(wt rs - ir rs - ir rs 1. The reflections which increase
the length of wt rs by exactly 1 are characterized by the following two propositions.
Proposition 1.2 ([17]). If w 2 S1 or B1 and only
if
and no k exists such that
Proposition 1.3. [1] If w 2 B1 , and i - j, then l(ws ij only if
and no k exists such that either of the following are true:
2. Main Results
In this section we give the definition of the vexillary elements in S n and B n . Then we
present the main theorem. The proof follows after several lemmas.
Definition 2. If w 2 S n then w is vexillary if
Similarly, if w then w is vexillary if
parts.
It follows from the definition of the Stanley symmetric functions (1.3) that if w is
vexillary then the number of reduced words for w is the number of standard tableaux of
a single shape (unshifted for w 2 S n or shifted for
For S n , this definition is equivalent to the original definition of vexillary given by
Lascoux and Sch-utzenberger in [14]. They showed that vexillary permutations w are
characterized by the condition that no subsequence a exists such that
. This property is usually referred to as 2143-avoiding. Lascoux and
Sch-utzenberger also showed that the Schubert polynomial of type A n indexed by w is a
flagged Schur function if and only if w is a vexillary permutation. One might ask if the
Schubert polynomials of type B, C or D indexed by a vexillary element could be written
in terms of a "flagged Schur Q-function."
Many other properties of permutations can be given in terms of pattern avoidance. For
example, the reduced words of 321-avoiding [3] permutations all have the same content,
and a Schubert variety in SL n =B is smooth if and only if it is indexed by a permutation
which avoids the patterns 3412 and 4231 [9]. Also, Julian West [24] and Simion
and Schmidt [19] have studied pattern avoidance more generally and given formulas for
computing the number of permutations which avoid combinations of patterns. Recently,
Stembridge [23] has described several properties of signed permutations in terms of pattern
avoidance as well.
We will define pattern avoidance in terms of the following function which flattens any
subsequence into a signed permutation.
VEXILLARY ELEMENTS IN THE HYPEROCTAHEDRAL GROUP 5
Definition 3. Given any sequence a 1 a of distinct non-zero real numbers, define
to be the unique element such that
ffl both a j and b j have the same sign.
ffl for all i; j, we have jb
For example, fl( - 6; 3; - 7; containing the subsequence
does not avoid the pattern - 41.
Theorem 1. An element w 2 B1 is vexillary if and only if every subsequence of length
4 in w flattens to a vexillary element in B 4 . In particular, w is vexillary if and only if it
avoids the following patterns:
This list of patterns was conjectured in [10]. Due to the large number of non-vexillary
patterns in (2.1) we have chosen to prove the theorem in two steps. First, we have verified
that the theorem holds for B 6 , see Lemma 2.1. Second, we show that any counter example
in B1 would imply a counter example in B 6 .
Lemma 2.1. Let w vexillary if and only if it does not contain any
subsequence of length 3 or 4 which flattens to a pattern in (2:1).
See the appendix for an outline of the code used to verify Lemma 2.1.
Lemma 2.2. Let w be any signed permutation. Suppose w
is a subsequence
of w and let
). Then the following statements hold:
1. If the last decent of w appears in position i r 2 fi then the last descent of
u will be in position r.
2. If in addition, w i s
and i s is the largest index in w such that this is true then
s is the largest index in u such that this is true.
3. If
then
and - jk are transpositions
of the same type.
One can check the facts above follow directly from the definition of the flatten function.
Lemma 2.3. For any v 2 B1 and any 0 there exists
an index k such that Similarly, if
there exists an index k such that either
6 SARA BILLEY AND TAO KAI LAM
Proof. If l(vt ir pick k such that v k is the largest value in fv k rg.
Then no j exists such that k
Say l(vs ir r such that v k chose k such that v k
is the largest value in fv k rg. Then no j exists such that k
1. On the other hand, if no such k exists, then choose
k such that v k is the smallest value in fv k ? \Gammav ig. Then no exists such
exists such that \Gammav r
Lemma 2.4. Given any w 2 B1 and any subsequence of w, say w
Similarly, if
Proof. If
so since the flatten map preserves the
relative order of the elements in the subsequence and signs. Therefore, l(vt jk
If
exists such that w i j
. This in
turn implies that no exists such that
and \Gammaw i k
so
since the flatten map preserves the relative order of the elements in the subsequence
and signs. Also, if
Therefore, l(vs jk
exists such that
, and no exists such that \Gammaw i k
. This in turn
implies that no exists such that \Gammav exists such that
Lemma 2.5. Given any w 2 B1 , if w is non-vexillary then w contains a subsequence
of length 4 which flattens to a non-vexillary element in B 4 .
Proof. Since w is non-vexillary then either Fw expands into multiple terms on the first
step of the recurrence in (1.5) or else non-vexillary. Assume
the first step of the recurrence gives
other terms
1g be an order preserving map onto the 4 smallest
distinct numbers in the range. Let w
fore, the recursion implies
Hence, w is not vexillary, and it follows that w contains the non-vexillary subsequence
If on the other hand the first step of the recursion gives rs - ir
and v is not vexillary. Assume, by induction on the number of steps until the recurrence
branches into multiple terms that v contains a non-vexillary subsequence say v a v b v c v d . If
VEXILLARY ELEMENTS IN THE HYPEROCTAHEDRAL GROUP 7
is exactly the same non-vexillary subsequence. So we
can assume the order of the set fa; b; c; d; than or equal to 6. Let
be an order preserving map which sends the numbers 1 through 6 to the 6 smallest distinct
integers in the range. Let w
contains a non-vexillary subsequence, hence v 0 is not vexillary by
Lemma 2.1. We will use the recursion on Fw 0 to show that w 0 is not vexillary in B 6 . From
Lemma 2.2 it follows that
By Lemma 2.3, rs
possibly other terms.
Irregardless of whether there are any other terms in expansion of Fw 0 , w 0 is not vexillary
since v 0 is not vexillary. Again by Lemma 2.1, this implies w 0 contains a non-vexillary
subsequence of length 4, say w
h . Hence, w contains the non-vexillary subsequence
This proves one direction of Theorem 1.
Lemma 2.6. Given any w 2 B1 , if w contains a subsequence of length 4 which flattens
to a non-vexillary element in B 4 then w is non-vexillary.
Proof. Assume w is vexillary then let w be the sequence of signed permutations
which arise in expanding
using the recurrence
(1.5). This recurrence terminates when the signed permutation w (k) is strictly increasing,
hence w (k) does not contain any of the patterns in (2.1). Replace w by the first w (i) such
that w (i) contains a non-vexillary subsequence and w (i+1) does not, and let
Say w a w b w c w d is a non-vexillary subsequence in w. If
would be exactly the same non-vexillary subsequence. This contradicts our choice of v.
So we can assume that the order of the set fa; b; c; d; than or equal to 6. As
in the proof of Lemma 2.5, let
be an order preserving map onto the smallest 6 distinct numbers in the range. Let
To simplify notation, we also
contains a
non-vexillary subsequence hence w 0 is not vexillary by Lemma 2.1. As in 2.5 one can
show
contains a non-vexillary subsequence and v 0 does not there must be another
term in Fw 0 indexed by a reflection - should
note that it is possible that i must be different types of
transpositions. 1.3 and the definition of the flatten
function, we have l(wt rs - jr rs ) ? 0. By Lemma 2.3 there exists a reflection - kr
such that l(wt rs - kr rs
We must have - kr 6= - ir since -
possibly other terms.
This proves w is not vexillary contrary to our assumption.
This completes the proof of Theorem 1.
The definition of vexillary can be extended to Stanley symmetric functions of type B
and D. These cover the remaining infinite families of root systems. For these cases, we
define vexillary to be the condition that the function is exactly one Schur P -function. The
signed permutations which are B and D vexillary can again be determined by avoiding
certain patterns of length 4.
Theorem 2. An element w 2 B1 is vexillary for type B if and only if every subsequence
of length 4 in w flattens to a vexillary element of type B in B 4 . In particular, w is vexillary
if and only if it avoids the following patterns:
An element w 2 D1 is vexillary for type D if and only if every subsequence of length
4 avoids the following patterns:
Note, that the patterns that are avoided by vexillary elements of type D are not all
type D signed permutations but instead include some elements with an odd number of
negative signs. The proof of Theorem 2 is very similar to the proof of Theorem 1 given
above. We omit the details in this abstract.
3. A rule for multiplication
Lascoux and Sch-utzenberger noticed that the transition equation for Schubert polynomials
of vexillary permutations can be used to multiply Schur functions [16][p.62].
L. Manivel asked if the transition equations for Schubert polynomials of types B, C, and
could lead to a rule for multiplying Schur Q-functions. The answer is "sometimes".
There are only certain shifted shapes - which can easily be multiplied by an arbitrary
Schur Q-function. Therefore, we have investigated a different problem. In this section
VEXILLARY ELEMENTS IN THE HYPEROCTAHEDRAL GROUP 9
we present an algorithm for multiplication of a Schur Q-function by a superfied Schur
Let OE be the homomorphism from the ring of symmetric functions onto the subring
generated by odd power sums defined by
The image of a Schur function under this map, OE(s - ), is called a superfied Schur function.
The superfied Schur functions appear in connection with the Lie super algebras [21][25].
The Stanley symmetric functions of type A and C which are indexed by permutations
are related via the superfication operator.
Proposition 3.1. [2][11][22] For v 2 S n , we have F
be any signed permutation. We denote the signed permutation
v. Also, if signed
permutation, let w \Theta v be [w
Lemma 3.2. For v 2 S1 and w 2 B1 we have
Proof. From (1.2), when v 2 S1 , F v is equal to F 1 n \Thetav since a 1 a only
assuming n is large enough, the reduced
words for w \Theta v are all shuffles of a reduced word for w with a reduced word for v. One
can check that the admissible monomials in F v\Thetaw are exactly the product of admissible
monomials in Fw and F 1 n \Thetav counted with their coefficients.
From Lemma 3.2 and Proposition 3.1 we have the following corollary.
Corollary 3.3. Let w 2 B1 such that
Then
and Fw\Thetav can be determined by the recursive formula in Proposition 1.1.
We remark that Corollary 3.3 can be used to multiply two Schur Q-functions in the
special case that one of the shapes is equivalent to a rectangle under jeu de taquin. In
this case, if a shifted shape - is equivalent to a rectangle ae then Q [12]. For each
straight shape ae one can easily choose a permutation v in S1 with that shape. However,
for any straight shape - other than a rectangle, the expansion of OE(s - ) will always be a
sum of more than one Q - . Since F only if v 2 S1 , the algorithm for
multiplying FwF v given above will not carry over for arbitrary elements of B1 .
4. The shape of a signed permutation
Given a vexillary element w, for which straight shape - does
for which shifted shape - does there are several
ways to determine this shape: the transition equation [16, p. 52], inserting a single
reduced word using the Edelman-Greene correspondence [4], or by rearranging the code
in decreasing order [20]. Similarly, for vexillary elements of type C one can find this
shape for the signed permutation by using the recursive formula (1.5) or by using the
Kra'skiewicz insertion [8] or Haiman procedures [7] on a single reduced word. There is
another method for computing the shape of a C n -vexillary element using jeu de taquin.
We describe this method below.
For any standard Young tableau U of shifted shape and any standard Young tableau
V of straight shape we form a new standard shifted tableau U   V by jeu de taquin as
follows:
1. Embed U into the shifted shape
2. Obtain a tableau R by filling the remaining boxes of ffi with starting from
the rightmost column and in each column from bottom to top.
3. Add j-j to each entry of V to obtain S.
4. Append R on the left side of S to obtain T .
5. Delete the box containing 1 0 in T . If the resulting tableau is not shifted, apply jeu
de taquin to fill in the box. Repeat the procedure for the box containing 2 0 and so
on until all the primed numbers are removed.
6. The resulting tableau of shifted shape is denoted U   V .
We illustrate the procedure with an example. Let
Then, Steps 1 through 4 will produce the following tableau:
Deleting the boxes and applying jeu de taquin as in Step 5 gives
U
Note that different choices for V will result in different shapes for U   V . If V is chosen
to be the the standard tableau with entries in the first row, in the
second row etc., we will say U   V has shape -   -. So, the result of combining - and -
by jeu de taquin in the example is the shape (4; 2)   (2;
There is a canonical decomposition of any signed permutation into the product of a
signed permutation and an unsigned permutation with nice properties. Let w be an
element of B n , not necessarily vexillary. Rearrange the numbers in w in increasing order
VEXILLARY ELEMENTS IN THE HYPEROCTAHEDRAL GROUP 11
and denote this new signed permutation by u. Let v 2 S n so that uv. Note that
and v is [4; 2;
Definition 4. Given an element w of S n , the code of w is defined to be the composition
g. The shape of w is defined to be the
transpose of the partition given by rearranging the code in decreasing order.
It is well known that if - is the shape of S n vexillary permutation v then G
Furthermore, for each standard tableau Q of shape - there exists a unique reduced word
for v with recording tableau Q under the Edelman-Greene correspondence [4]. Also, the
reduced words for u are in bijective correspondence with the shifted standard tableau of
shape -.
Recall from Proposition 1.1 that if u is a strictly increasing signed permutation, we
have F is the strictly decreasing sequence given by fju
is the same set as fjw 0g. Therefore, it is easy to determine the shape of u.
Definition 5. For any w 2 B n , let and u is
strictly increasing. Let - be the shape of u, and let - be the shape of v as an element of
. Define the shape of w , denoted -(w), to be the shape -   -.
In the case of vexillary signed permutations, we claim that is the
shape of w. This is true in the case when w has all its numbers in increasing order.
Before we prove the claim, here are some results that we will need. We refer the reader
to the references for their proofs.
Proposition 4.1. [2][11][22] Let w be a signed permutation with no positive numbers.
-. The shape of w is
Proposition 4.2. [11, Theorem 3.24] Let a 1 a 2 a am be a reduced word for some signed
permutation w in B n . Suppose Q and Q 0 are the recording tableaux that are obtained by
applying Kra'skiewicz insertion on a 1 a 2 a am and a 2 a
also be obtained by deleting the box containing the entry 1 in Q and applying jeu de taquin
to turn Q into a shifted tableau and subtracting 1 from every entry.
Theorem 3. For , the shape of w, -(w) is the shape of some Kra'skiewicz recording
tableau for some reduced word of w. Hence Q -(w) appears in the expansion of Fw with
a non-zero coefficient.
Proof. Let uv be the canonical decomposition of w. Given
a reduced word a, denote the Kra'skiewicz recording tableau by K(a). Fix a reduced
word b of u with recording tableau U = K(b) under Kra'skiewicz insertion. Let V be
the standard tableau of straight shape with entries filled sequentially in rows from left
to right, top to bottom. Let c 2 R(v) be the reduced word which inserts under the
Edelman-Greene correspondence to V . We will actually show that the reduced word
rise to the Kra'skiewicz recording tableau U   V which by definition has
the same shape as -(w).
We prove the claim by induction on the number of positive numbers in w. When there
is no positive number in w, this follows from Proposition 4.1 and there are no jeu de
taquin slides necessary.
Now suppose w has p positive numbers. Let be the
signed permutation that is the same as w except that m is signed. We can write w 0 as a
product is the arrangement of w 0 in increasing order and v is the same as in
the decomposition for w. Let
Note that u
with a be a reduced word for
z, then abc 2 R(w 0 ). Let U 0 be the recording tableau for ab 2 R(u 0 ).
positive numbers, by the induction hypothesis, the recording tableau
of the Kra'skiewicz insertion of abc 2 R(w 0 ) is given by U 0   V .
From Proposition 4.2, the tableau U can be obtained from U 0 by deleting m boxes
labeled applying jeu de taquin to fill them up. Since u and u 0 are strictly
increasing, by Proposition 1.1, we know their shapes explicitly. Note that u 0 was chosen
so that the m boxes which are vacated from the shape for u 0 in the jeu de taquin process
form a vertical strip in the (n \Gamma p)th column (since w 2 B n ). Therefore, filling these m
boxes with the next higher primed numbers we obtain the tableau T which appears in
Step 4 when computing -(w). Therefore, since U 0   V gives K(abc) and evacuation of a
from U 0 gives U then U   V must give K(bc). Therefore, U   V is the recording tableau
for bc 2 R(w).
From this proof one can also prove the following corollaries.
Corollary 4.3. For any vexillary element w 2 B n , we have
where -(w) is the shape of w as in Definition 5.
Corollary 4.4. Given any w 2 B n , let uv be the canonical
decomposition of w. For any a 2 R(u) and any b 2 R(v) with recording tableaux U and
respectively, then U   V is the recording tableau for the reduced word ab 2 R(w).
5. Open Problems
The vexillary permutations in S n have many interesting properties. We would like to
explore the possibility that these properties have analogs for the vexillary elements in
1. Is there a relationship between smooth Schubert varieties in SO(2n + 1)=B and
vexillary elements? In particular, does smooth imply vexillary as in the case of S n ?
2. Is there a way to define flagged Schur Q-functions so that the Schubert polynomial
indexed by w of type B or C is a flagged Schur Q-functions if and only if w is
vexillary?
VEXILLARY ELEMENTS IN THE HYPEROCTAHEDRAL GROUP 13
3. Are there other possible ways to define vexillary elements in B n so that any of the
above questions can be answered?
Below is a portion of the LISP code used to verify Theorem 1 for B 6 . The calculation
was done on a Sparc 1 by running (grind-patterns 6 'c).
(setf *avoid-patterns*
(list
(defun grind-patterns (n type)
(flet ((helper (perm)
(if (not (eq (and (avoid-subsequences perm
(avoid-subsequences perm 4))
(vex-p perm type)))
(format t "ERROR::NEW PATTERN: ~a ~%" perm)
(format t "."))))
(all-perm-tester n #'helper
(defun avoid-subsequences (the-list size)
(let ((results t))
(catch 'foo
(flet ((helper (tail)
(when (member (flatten-seq tail) *avoid-patterns*
:test #'equal)
(setf results nil)
(throw 'foo nil)
(all-subsequences-tester (reverse the-list) size #'helper nil))
(throw 'foo t))



--R

Transition Equations for Isotropic Flag Manifolds
Schubert polynomials for the classical groups
Some Combinatorial Properties of Schubert Polyno- mials

Schubert Poly- nomials
Schubert Polynomials and the NilCoxeter Algebra
Dual equivalence with applications

Criterion for smoothness of schubert varieties in SL(n)
Bn Stanley


Structure de hopf de l'anneau de cohomologie et de l'anneau de grothendieck d'une variete de drapeaux

Oxford University Press

The geometry of flag manifolds
Algebraic Combin.
of Combinatorics
On the number of reduced decompositions of elements of Coxeter groups

personal communication

Permutations with forbidden sequences
A theory of shifted Young tableaux
--TR

--CTR
Bridget Eileen Tenner, On expected factors in reduced decompositions in type B, European Journal of Combinatorics, v.28 n.4, p.1144-1151, May, 2007
S. Egge , Toufik Mansour, 132-avoiding two-stack sortable permutations, Fibonacci numbers, and Pell numbers, Discrete Applied Mathematics, v.143 n.1-3, p.72-83, September 2004
stanley symmetric function;reduced word;hyperoctahedral group;vexillary