Abacus-tournament Models of Hall-Littlewood Polynomials

Transcription

1 Abacus-tournament Models of Hall-Littlewood Polynomials Andrew J. Wills Dissertation submitted to the Faculty of the Virginia Polytechnic Institute and State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics Nicholas A. Loehr, Chair Ezra A. Brown William J. Floyd Peter A. Linnell December 7, 05 Blacksburg, Virginia Keywords: Symmetric polynomials, Hall-Littlewood polynomials, abacus-tournaments, Pieri rules. Copyright 05, Andrew J. Wills

2 Abacus-tournament Models of Hall-Littlewood Polynomials Andrew J. Wills (ABSTRACT) In this dissertation, we introduce combinatorial interpretations for three types of Hall- Littlewood polynomials (denoted R, P, and Q ) by using weighted combinatorial objects called abacus-tournaments. We then apply these models to give combinatorial proofs of properties of Hall-Littlewood polynomials. For example, we show why various specializations of Hall-Littlewood polynomials produce the Schur symmetric polynomials, the elementary symmetric polynomials, or the t-analogue of factorials. With the abacus-tournament model, we give a bijective proof of a Pieri rule for Hall-Littlewood polynomials that gives the P -expansion of the product of a Hall-Littlewood polynomial P µ with an elementary symmetric polynomial e k. We also give a bijective proof of certain cases of a second Pieri rule that gives the P -expansion of the product of a Hall-Littlewood polynomial P µ with another Hall-Littlewood polynomial Q (r). In general, proofs using abacus-tournaments focus on canceling abacus-tournaments and then finding weight-preserving bijections between the sets of uncanceled abacus-tournaments.

3 Contents Introduction Background 5. Partitions Permutations Quantum Factorials and Binomial Coefficients Symmetric Polynomials Monomial Symmetric Polynomials Elementary Symmetric Polynomials Complete Symmetric Polynomials Schur Polynomials Monomial Antisymmetric Polynomials Pieri Rules for Schur Polynomials Hall-Littlewood Polynomials Combinatorial Interpretations of Hall-Littlewood Polynomials 3. Abacus-Tournaments A Combinatorial Interpretation of a δ(n) R Blocks and Involutions Global and Local Exponent Collisions Cancellation Theorem for Local Exponent Collisions Leading Abacus-Tournaments iii

4 3.7 A Combinatorial Explanation of Division by t-factorials A Combinatorial Interpretation for a δ(n) P A Combinatorial Interpretation for a δ(n) Q Local Exponent Collision Lemmas The Single Gap Collision Lemma The Single Bead Collision Lemma Specializations of Hall-Littlewood Polynomials Specialization at t = Specialization at t = Specialization at x N+ = Specializations at Partitions with One Column Specializations at Partitions with One Row A Pieri Rule for Hall-Littlewood Polynomials A Combinatorial Model for the Right Side of the Pieri Rule Combinatorial Models for a δ(n) P µ e k A Bijection Between the Two Models A Second Pieri Rule for Hall-Littlewood Polynomials A Combinatorial Model for Q (r,0 N ) Subproblem P Subproblem P A Combinatorial Model for a δ(n) P µ Q (r,0 N ) A Combinatorial Model for the Right Side of the Pieri Rule A Bijection Between the Two Models Subproblem P A Combinatorial Model for a δ(n) P µ Q (r,0 N ) A Combinatorial Model for the Right Side of the Pieri Rule iv

5 7.4.3 A Bijection Between the Two Models Subproblem P v

6 List of Figures. An example of an abacus-tournament An example of an abacus-tournament The square of block {,, 3, 4, 5} in an abacus-tournament The squares of the -blocks in an abacus-tournament An abacus-tournament to pair with the one in Figure An abacus-tournament to pair with the one in Figure An abacus-tournament for = (3 3, 4, 0) that is leading in B An abacus-tournament for = (3 3, 4, 0) with a permuted abacus An abacus-tournament for = (3 3, 4, 0) with local outdegrees in B out of order An abacus-tournament for = (3 3, 4, 0) with local outdegrees in B in order An abacus-tournament that is leading in Pos (0) An example of a shaded abacus-tournament x and x 8 have equal local exponents in blocks D and B C An abacus-tournament A such that upset(a) is not right-justified An abacus-tournament A such that upset(a ) is right-justified and upset(a ) An abacus-tournament A such that upset(a ) is right-justified and upset(a ) < An abacus-tournament without upsets An abacus-tournament in the set X vi

7 5.3 Bead 9 and the edges in row are removed from Figure 5. by J An abacus-tournament with shape = ( 3, 0 4 ) The abacus J(A ) with position set µ = ( 3, 0 ) and word An abacus-tournament in LAT B /µ An abacus-tournament in LAT B /µ and LAT Bµ An abacus-tournament in the domain of I An abacus-tournament that cancels with the one in Figure An abacus-tournament in JAT (7,0 5 ) Abacus-tournament R(A, T ) JAT (7,0 5 ) Outlines for µ and meeting Subproblem P restrictions An abacus-tournament A LAT Bµ µ An abacus-tournament A LAT Bµ µ An abacus-tournament A LAT Bµ µ An abacus-tournament in JAT µ () An abacus-tournament in JAT µ () An abacus-tournament A JAT µ () The abacus-tournament A where R(A, T ) = A An abacus-tournament A such that A X but A LAT B An abacus-tournament A JAT µ () An abacus-tournament A LAT Bµ µ The abacus-tournament R(A, T ) The abacus-tournament R(A, S ) The abacus-tournament A such that R (A ) = (A, T ) An abacus-tournament A LAT Bµ µ The abacus-tournament R(A, T ) vii

8 List of Tables. Content monomials for SSYT 3 ((,, 0)) Signs and weights for the set LAbc((3,, )) The signed weights of abacus-tournaments for = (, 0) The abacus-tournaments for = ( 3, 0) leading in B and their signed weights The shaded abacus-tournaments for = (, 0) viii

9 Chapter Introduction Symmetric polynomials are polynomials in multiple variables that are unaffected if any two of the variables are interchanged. For background information on symmetric polynomials, see Sagan [], Stanley [3], and Loehr [9]. Schur functions are a well-known type of symmetric polynomial that have connections to tableau enumeration and the representation theory of the symmetric group. Many transition matrices between different bases of the vector space of symmetric polynomials have combinatorial interpretations (see []). For example, Eğecioğlu and Remmel [4] described the transition matrix between Schur polynomials s and monomial symmetric polynomials m µ, called the inverse Kostka matrix, with signed rim hook tabloids. It is desirable to find combinatorial proofs of Schur function identities. Loehr used objects called labeled abaci to give bijective proofs of antisymmetrized versions of many Schur polynomial identities in [8]. Included were Pieri rules and the Littlewood- Richardson rule for Schur polynomials, which describe the Schur expansions of products of Schur polynomials with various other symmetric polynomials. Hall-Littlewood polynomials (which come in three types, denoted R, P, and Q ) are an important basis of symmetric polynomials in N variables. These polynomials have a parameter t and are indexed by partitions of precisely N nonnegative parts. A generalization of Schur polynomials and monomial symmetric polynomials, Hall-Littlewood polynomials were first defined in 96 by Littlewood [7] based on work by Philip Hall [6] studying the lattice structure of finite abelian p-groups. Since then, they have been extensively studied by Macdonald [, III] and others from a predominantly algebraic standpoint. As with Schur polynomials, there are combinatorial descriptions of Hall-Littlewood polynomial transition matrices. Loehr, Serrano, and Warrington studied some of these using starred semistandard tableaux in [0], and Carbonara used special tournament matrices to describe the transition matrix between P and Schur polynomials []. There are also many algebraic identities for Hall-Littlewood polynomials; see Macdonald [] for a thorough algebraic treatment of these identities. Since Schur polynomials are closely related to Hall-Littlewood polynomials, there are identities for Hall-Littlewood polynomials analogous to many of the

10 Andrew J. Wills Chapter. Introduction Figure.: An example of an abacus-tournament identities in Loehr s paper proving Schur function identities with abaci [8]. Consequently, Loehr s paper provides a motivating framework suggesting that certain Hall-Littlewood polynomial identities may have analogous bijective proofs. In this dissertation, we introduce combinatorial objects called abacus-tournaments. Each abacus-tournament has three associated components: a partition, a labeled abacus, and a tournament. Figure. shows a visual depiction of one abacus-tournament for the partition = (3, 3, 3,, 0, 0, 0, 0). Each abacus-tournament has an associated monomial in several variables, called its signed weight, that is calculated from various aspects of the abacustournament. When the signed weights of a specific set of abacus-tournaments, all sharing the same partition, are summed together, the resulting polynomial is exactly a δ(n) R, the product of the Vandermonde determinant a δ(n) and the Hall-Littlewood polynomial R. Then, giving a bijective explanation for the divisibility of R by products of t-factorials results in combinatorial models for a δ(n) P and a δ(n) Q. We find that proofs with abacustournaments, which incorporate abaci and tournaments, have significantly more complicated interactions between objects than proofs with labeled abaci alone. Our strategy for proving identities for Hall-Littlewood polynomials is to first antisymmetrize (multiply by a δ(n) ) and then compare the abacus-tournament models for the two sides of the identity. Such a comparison is formally established by demonstrating a weight-preserving bijection between the two sets of abacus-tournaments, and this ensures that the polynomials represented by the two sets are equal. Frequently, such a bijection is impossible to find between an initial pair of sets. In this case, one or both models must be cleared of extraneous

11 Andrew J. Wills Chapter. Introduction 3 objects by pairing off abacus-tournaments that have opposing signed weights using signreversing involutions. These pairs of objects cancel out in the sum of signed weights, leaving two core sets of abacus-tournaments that can be matched by a bijection. The identities for Hall-Littlewood polynomials that we prove in this dissertation fall into two categories. The first type of identity relates Hall-Littlewood polynomials to other symmetric polynomials. When evaluated at specific instances of partitions or variables x,..., x N, and t, Hall-Littlewood polynomials specialize to other symmetric polynomials, or may produce a sum of symmetric polynomials. In proofs of this type of identity, we draw on other established combinatorial models for symmetric polynomials to pair with the abacus-tournament model. A list of some of the identities that we prove combinatorially follows. See Chapter or Macdonald [] for the definitions of notation used here. R (0) (x,..., x N ; t) = [N]! t, the t-analogue of N factorial. P (x,..., x N ; 0) = s (x,..., x N ), where s denotes the Schur symmetric polynomial. P ( r,0 N r )(x,..., x N ; t) = e r (x,..., x N ), where e r denotes the rth elementary symmetric polynomial. P (x,..., x N, 0; t) = P (x,..., x N ; t). On the left side, we have set x N+ = 0. P (r,0 N )(x,..., x N ; t) = r k=0 ( t)k s (r k, k,0 N k )(x,..., x N ). The second type of identity found in this dissertation advances our understanding of how Hall-Littlewood polynomials behave in the algebra of symmetric polynomials. It is desirable to prove Pieri rules for Hall-Littlewood polynomials that express the product of a Hall- Littlewood polynomial P µ with some symmetric polynomial f as a sum of Hall-Littlewood polynomials P. To this end, we give a bijective proof of a Pieri rule that describes how to express P µ e r, the product of the Hall-Littlewood polynomial P µ and the elementary symmetric polynomial e r, as a sum of Hall-Littlewood polynomials P. We also give a bijective proof of certain cases of a second Pieri rule to describe the product of P and Q (r). The rest of this dissertation is organized as follows. Chapter provides the necessary definitions and notation involving symmetric polynomials, antisymmetric polynomials, and Hall- Littlewood polynomials. Chapter also introduces abaci and tournaments, which are key ingredients in our combinatorial models for Hall-Littlewood polynomials. Chapter 3 establishes abacus-tournament models for the antisymmetrized Hall-Littlewood polynomials a δ(n) R, a δ(n) P, and a δ(n) Q. We also explain combinatorially why R is divisible by products of t-factorials. Chapter 4 provides two mechanisms for canceling sets of abacus-tournaments: the Single Gap Collision Lemma and the Single Bead Collision Lemma. Chapter 5 is devoted to proving identities relating specialized Hall-Littlewood polynomials to many of the symmetric polynomials defined in Chapter. Chapter 6 gives a bijective proof of the first Pieri rule for Hall-Littlewood polynomials. Chapter 7 gives bijective proofs of some special

12 Andrew J. Wills Chapter. Introduction 4 cases of the second Pieri rule for Hall-Littlewood polynomials and discusses future research to prove the general form of this rule.

13 Chapter Background We now present preliminary material and establish notation in order to introduce the reader to a number of classic symmetric polynomials, culminating in Hall-Littlewood polynomials. Along the way, we define three important combinatorial objects: tournaments, tableaux, and labeled abaci. We also present combinatorial interpretations for two Pieri rules for Schur polynomials, which inspire Hall-Littlewood polynomial versions of these rules discussed in Chapters 6 and 7 of this dissertation.. Partitions Definition. A partition of an integer k is a weakly decreasing sequence = (,,..., N ) of N nonnegative integers with N = k. Informally, a partition breaks up an integer k into integer parts,,..., N. In general, we may be interested in partitions with a particular number of parts N, or partitions of a particular integer k. Definition. Define l() to be the number of nonzero parts of a partition. The size of a partition is = N. Define Par(k) to be the set of partitions of size k, define Par N to be the set of partitions with exactly N nonnegative parts, and define Par N (k) to be the set of partitions of k with exactly N nonnegative parts. For Par N and i 0, let m i () denote the number of parts of of size i. We can append N l() parts of size zero to a partition with fewer than N nonzero parts to obtain a partition in Par N ; then l() + m 0 () = N for such. 5

14 Andrew J. Wills Chapter. Background 6 Example 3. The partition = (4,,, 0) Par 4 (8) is a partition of 8 with 4 parts. An alternative notation for combines parts of equal size into a single term with an exponent. In this case, we can write = (4,, 0). This partition has one part of size 4, two parts of size, and one part of size 0. Definition 4. For Par N (k), the Ferrers diagram of is dg() = {(i, j) N N : i N, j i }. We can visually represent dg() as an array of N left-justified rows having k cells such that row i contains precisely i boxes for i N. Rows with no boxes are marked with a single vertical line. Example 5. We can depict = (4,, 0) by left-justifying four rows of boxes where the first row has 4 boxes, the second and third rows have boxes, and the fourth row has no boxes: dg() = Definition 6. For = (,..., N ) Par N and all i 0, define i = {j : j i}. Note that i is the number of cells in the ith column of the diagram of. Also, 0 = N, i = 0 for all i >, and m i () = i i+ for all i 0. Example 7. For = (4,, 0), 0 = 4, = 3 =, and 3 = = 4. Definition 8. If, µ Par N are partitions such that i µ i for all i, define the skew shape /µ = {(i, j) : i N, µ i < j i }. The skew shape /µ can be obtained visually from the diagram of by overlaying dg(µ) and erasing any overlapping squares. If µ = (0 N ) is the zero partition, then /µ = dg(). Example 9. If = (4,, 0) and µ = (,, 0), then /µ =

15 Andrew J. Wills Chapter. Background 7 On the other hand, (5,,, 0)/(3,,, 0 ) = Definition 0. A skew shape /µ, where, µ Par N, is a vertical r-strip if /µ has precisely r cells and each row has at most one cell. Similarly, /µ is a horizontal r-strip if /µ has precisely r cells and each column has at most one cell. If µ Par N, let V(µ, r) = { Par N : /µ is a vertical r-strip}, and H(µ, r) = { Par N : /µ is a horizontal r-strip}. Example. The skew shape (5,,, 0)/(3,,, 0 ), displayed in the previous example, is a horizontal 4-strip and is not a vertical strip of any size. The skew shape /µ for = (4,, 0) and µ = (,, 0), also displayed in the previous example, is neither a vertical 4-strip nor a horizontal 4-strip. The following skew shape is a vertical 5-strip but not a horizontal strip. (5, 4, 3,, 0)/(4, 3,, 0 ) = Example. If µ = (3,, 0) Par 4, then V(µ, 3) = {(3 3, ), (4, 3,, ), (4, 3, 0)}. We display the partitions V(µ, 3) below with s in cells which overlap with µ. The cells without s form the skew shape /µ. Similarly, H(µ, 3) = {(3, ), (4, 3 ), (4, 3,, ), (5,, ), (5, 3,, 0), (6,, 0)},

16 Andrew J. Wills Chapter. Background 8 which we display below. Note that (4, 3,, ) is an partition of both V(µ, 3) and H(µ, 3).. Permutations Definition 3. A word of length N over {,,..., N} is a sequence w = w w w N where each w i {,..., N}. Such a word is also a permutation of {,..., N} provided each element of {,..., N} appears in w exactly once. Equivalently, we can define a word w w w N of length N over {,..., N} as a function w : {,..., N} {,..., N} such that w(i) = w i for all i. In this case, we say w is a permutation of {,..., N} iff w is a bijective function. Let S N denote the set of permutations of {,..., N}. Example 4. Both u = 6354 and v = 635 are words of length 6. The word u is in S 6, while v is not. As a function, u maps to 6, to, 3 to, and so on. Example 5. The set of permutations S 3 is S 3 = {3, 3, 3, 3, 3, 3}. Definition 6. Let w = w w w N S N be a permutation. An inversion of w is a pair (i, j) such that i < j and w i > w j. Let inv(w) denote the number of inversions of w and define the sign of w to be sgn(w) = ( ) inv(w). Example 7. For N = 6, the permutation u = 6354 has 6 total inversions: (6, ), (6, ), (6, 3), (6, 4), (6, 5), (5, 4). For more details on permutations (e.g., group structure, inverses, and compositions) see [9, Ch. 9].

17 Andrew J. Wills Chapter. Background 9.3 Quantum Factorials and Binomial Coefficients Definition 8. For a variable t and positive integer k, define Furthermore, define [0] t = 0, [0]! t =, and We call [k]! t the quantum factorial of k. [k] t = + t + t + + t k = tk t. [k]! t = [] t [] t [k] t. Example 9. To find the quantum factorial of 3, we compute [3]! t = [] t [] t [3] t = ( + t) ( + t + t ) = + t + t + t 3. The next theorem shows that the coefficients of monomials of the form t b in [N] t count objects: namely, the number of permutations in S N with precisely b inversions. Theorem 0. For N, w S N t inv(w) = [N]! t. A proof of this theorem can be found in [9, Thm. 6.6]. Definition. For a variable t and integers k, n with 0 k n, define [ ] [ ] n n [n]! t = =. k t k, n k t [k]! t [n k]! t We call [ ] n a quantum binomial coefficient. k t Example. The quantum binomial coefficient for n = 4 and k = is [ ] 4 = [4]! t t []! t []! t = ( + t)( + t + t )( + t + t + t 3 ) ( + t)( + t) = + t + t + t 3 + t 4

18 Andrew J. Wills Chapter. Background 0.4 Symmetric Polynomials We now define what it means for a multivariable polynomial to be symmetric, and we define a few examples of symmetric polynomials. The set of symmetric polynomials in a fixed number of variables form a vector space, and some of the symmetric polynomials provide specific bases for this vector space. See [3] for algebra notation not defined here. Definition 3. For β N N, let x β = x β x β x β N N. We say x β is a monomial of degree β = β + β + + β N. Definition 4. For a permutation w S N and a polynomial f K[x, x,..., x N ], define the action w f by w f(x, x,..., x N ) = f(x w(), x w(),..., x w(n) ). Furthermore, if h = f/g is the quotient of two polynomials f, g K[x, x,..., x N ], define w h(x, x,..., x N ) = w f(x, x,..., x N ) w g(x, x,..., x N ). Definition 5. For a field K, a polynomial f in K[x, x,..., x N ] is symmetric iff w f(x, x,..., x N ) = f(x, x,..., x N ) for all w in the symmetric group S N. A polynomial f is homogeneous of degree k if every monomial x β appearing in f with nonzero coefficient has degree k. The 0 polynomial is considered to be homogeneous of every degree. Example 6. Let N = 3 and K = Q. The polynomial f(x, x, x 3 ) = x x x x x 0 x 3 + x x x x 0 x x 3 + x x 0 x 3 + x 0 x x 3 is symmetric. For example, if w = 3, then w f(x, x, x 3 ) = f(x, x, x 3 ). In particular, the term x x 0 x 3 maps to x x x 0 3 in w f(x, x, x 3 ). On the other hand, the polynomial is not symmetric because, for example, g(x, x, x 3 ) = x 3 x x 3 + x x 3 x 3 w g(x, x, x 3 ) = x x 3 x 3 + x x x 3 3 g(x, x, x 3 ). Note that f is homogeneous of degree 3, and g is homogeneous of degree 5. Definition 7. Let Λ N be the set of all symmetric polynomials in K[x,..., x N ]. For all k 0, let Λ k N be the set of polynomials in Λ N that are homogeneous of degree k.

19 Andrew J. Wills Chapter. Background Here and below, we will fix an integer N that determines both the number of variables x,..., x N for a multivariable polynomial and the number of parts in a partition Par N including parts of size zero. If a polynomial is displayed without a list of variables, the reader may assume such a polynomial involves variables x,..., x N unless otherwise specified. Theorem 8. Λ N and Λ k N are K-vector spaces and Λ N = k 0 Λ k N. See [9, p. 386] for a proof of this theorem..5 Monomial Symmetric Polynomials Definition 9. Given a sequence β N N, define sort(β) N N by sorting the entries of β into weakly decreasing order. For Par N, let M() = {β N N : sort(β) = } denote the set of sequences, or exponent vectors, that sort to the particular partition. A symmetric polynomial in N variables indexed by a partition can be created by summing every monomial x β such that its exponent sequence β sorts to. Definition 30. For a partition Par N and variables x,..., x N, the monomial symmetric polynomial indexed by is m (x, x,..., x N ) = x β = x β x β x β N N. β M() Example 3. For N = 3 and the partition = (,, 0), β M() m (,,0) (x, x, x 3 ) = x x x x x 0 x 3 + x x x x 0 x x 3 + x x 0 x 3 + x 0 x x 3. For N = 3 and the partition = (3,, ), m (3,,) (x, x, x 3 ) = x 3 x x 3 + x x 3 x 3 + x x x 3 3. Note that m (,,0) and m (3,,) are symmetric polynomials that are homogeneous of degree 3 and 4, respectively. Theorem 3. For all Par N, the monomial symmetric polynomial m is a homogeneous symmetric polynomial of degree. For every k 0 and N, {m : Par N (k)} is a basis for Λ k N. See [9, Thm. 0.9] for a proof of this theorem.

20 Andrew J. Wills Chapter. Background.6 Elementary Symmetric Polynomials The next polynomials are another example of homogeneous symmetric polynomials. Definition 33. For an integer r such that r N, the rth elementary symmetric polynomial in N variables is e r (x,..., x N ) = x i x i x ir = ( ) x j. Example 34. For N = 4 and r = 3, i <i < <i r N S {,...,N}: S =r e 3 (x, x, x 3, x 4 ) = x x x 3 + x x x 4 + x x 3 x 4 + x x 3 x 4. Theorem 35. For all r such that r N, the rth elementary symmetric polynomial is a homogeneous symmetric polynomial of degree r. See [9, Sec. 0.4] to prove this theorem. j S.7 Complete Symmetric Polynomials Definition 36. For an integer r, the rth complete symmetric polynomial in N variables is h r (x,..., x N ) = x i x i x ik. Example 37. For N = 4 and r =, i i i r N h (x, x, x 3, x 4 ) = x + x + x 3 + x 4 + x x + x x 3 + x x 4 + x x 3 + x x 4 + x 3 x 4. Theorem 38. For all r, the rth complete symmetric polynomial is a homogeneous symmetric polynomial of degree r. See [9, Sec. 0.4] to prove this theorem..8 Schur Polynomials Definition 39. For a partition, a tableau of shape is a function T : dg() N +.

21 Andrew J. Wills Chapter. Background 3 A tableau of shape can be displayed by filling in boxes of the Ferrers diagram of by placing the value T (i, j) in the box in row i and column j. Each box, corresponding to an ordered pair (i, j) dg(), is called a cell of T. For example, R = S = T = are tableaux of shape (4,,, 0). Definition 40. A tableau T of shape is semistandard if the values in each row weakly increase from left to right and the values in each column strictly increase from top to bottom. For a partition Par N, let SSYT N () denote the set of all semistandard tableaux of shape taking values in {,..., N}. A semistandard tableau T of shape is standard if T is a bijection from dg() to {,..., }, i.e., each number from to appears exactly once in T. In the example above, S is semistandard, T is standard, and R is neither. Example 4. The set of semistandard tableaux for the partition = (, ) Par is { } SSYT ((, )) =,. Neither of these are standard tableaux. On the other hand, the set of semistandard tableaux for the partition = (,, 0) Par 3 is SSYT 3 ((,, 0)) =, 3, 3,, 3,, 3, 3. The fifth and sixth tableaux are also standard tableaux. Each monomial symmetric polynomial m is a sum of monomials x β arising from objects β M(). Similarly, we can build symmetric polynomials called Schur polynomials by adding up certain monomials indexed by semistandard tableaux. Definition 4. The content of a tableau T of shape is c(t ) = (c, c,... ) where c k = {(i, j) dg() : T ((i, j)) = k}. Informally, c k is the number of times k appears in T. Given variables x, x,..., the content monomial of T is x c(t ) = x c x c x c k k.

22 Andrew J. Wills Chapter. Background 4 Example 43. The tableau S displayed below has one, zero s, two 3 s, and so on, so c(s) = (, 0,,, 0, 0, 0,,, 0, 0,... ). Therefore, x c(t ) = x x 3x 4x 8x 9. S = Example 44. Table. shows the content monomials for each semistandard tableau in SSYT 3 ((,, 0)). When summed together, these monomials form the polynomial x x + x x 3 + x x 3 + x x + x x x 3 + x x 3 + x x 3, which is a symmetric polynomial in 3 variables. The next definition generalizes this example. Table.: Content monomials for SSYT 3 ((,, 0)). T SSYT 3 ((,, 0)) x c(t ) T SSYT 3 ((,, 0)) x c(t ) x x 3 x x x x x 3 x x x x x 3 3 x x 3 x x 3 3 x x 3 Definition 45. For a partition Par N, the Schur polynomial in N variables indexed by is s (x,..., x N ) = x c(t ). T SSYT N ()

23 Andrew J. Wills Chapter. Background 5 Example 46. We calculated in the previous example that s (,,0) = x x + x x 3 + x x 3 + x x + x x x 3 + x x 3 + x x 3. From Example 4, we see that s (,) = x x + x x. Theorem 47. For a partition Par N, s (x,..., x N ) is a symmetric polynomial that is homogeneous of degree. Furthermore, for fixed k 0 and N, {s (x,..., x N ) : Par N (k)} is a basis for Λ k N. See [9, Thm. 0.49] for a proof..9 Monomial Antisymmetric Polynomials The next type of polynomial we need is called the monomial antisymmetric polynomial. These polynomials have both an algebraic definition and a combinatorial interpretation. As the name suggests, monomial antisymmetric polynomials are antisymmetric rather than symmetric. Definition 48. A polynomial f in N variables x,..., x N is antisymmetric iff for all w S N. f(x w(),..., x w(n) ) = sgn(w)f(x,..., x N ) Definition 49. For N, define δ(n) = (N, N,...,,, 0) Par N. We are about to define monomial antisymmetric polynomials a µ (x,..., x N ) where µ is a partition with N distinct parts. There is a one-to-one correspondence between partitions Par N and partitions µ Par N such that µ has distinct parts, given by µ = +δ(n). Consequently, we often index a monomial antisymmetric polynomial for µ as a +δ(n) where µ = + δ(n). Definition 50. For partitions, µ Par N such that µ = + δ(n) and variables x,..., x N, define a µ (x,..., x N ) = a +δ(n) (x,..., x N ) = N sgn(w) x µ i w(i). w S N i= Example 5. Let N = 3 and = (3,, ). Then + δ(3) = (5, 3, ) and a (5,3,) (x, x, x 3 ) = x 5 x 3 x 3 + x 3 x x x x 5 x 3 3 x 3 x 5 x 3 x 5 x x 3 3 x x 3 x 5 3. Theorem 5. The monomial antisymmetric polynomials a +δ(n) are antisymmetric polynomials.

24 Andrew J. Wills Chapter. Background 6 See [9, Def. 0.8] for details of a proof. As with the previously studied polynomials, individual monomials in the expression for a monomial antisymmetric polynomial can be obtained by calculating an exponent vector from a combinatorial object. In this case, the objects under consideration are called abaci (see [9, Ch. ], [8]). Definition 53. A labeled abacus with N beads is an ordered pair (, v) such that Par N and v S N. Let LAbc denote the set of all labeled abaci and, for Par N, define LAbc() = {(, v) : v S N }. To display an abacus (, v), place beads on a horizontal line, called the bead runner, in positions given by pos(, v) = + δ(n). By Theorem??, µ = pos(, v) = (µ > µ > > µ N ) Par N is a set of distinct positions. Positions on the bead runner not found in pos(, v) are marked on the runner with bead gaps. Bead positions start at 0 and are listed from left to right. The beads are then labeled with integers given by the word v = v v v N from right to left. If pos(, v) = (µ > µ > > µ N ) and w(, v) = v v v N, the ith bead from the right in (, v) is located in position µ i and is labeled v i. Example 54. Let N = 8, = (3 3, 4, 0), and v = Then pos(, v) = (0, 9, 8, 6, 5, 4, 3, 0). The labeled abacus (, v) is drawn below In proofs involving abaci, it is common to produce a new abacus by permuting the positions {,..., N} of the labels in w(, v) = v v v N. We typically write permutations of positions in cycle notation and abacus words in one-line form (see [3]). Example 55. As in the previous example, let N = 8, = (3 3, 4, 0), and v = Let a = (, 3, 4, 5)(, 6, 7)(8) permute positions in w(, v). The new abacus (, u) with position set pos(, u) = pos(, v) and word u = v a = is drawn below We can assign a sign and a weight monomial to each labeled abacus (, v) as follows. Definition 56. Given a labeled abacus (, v) with N beads and µ = pos(, v) = + δ(n), define the weight of (, v) to be N wt(, v) = x µ i v i. i=

25 Andrew J. Wills Chapter. Background 7 Define the sign of (, v) to be sgn(, v) = sgn(v) = ( ) inv(v). Informally, a bead labeled j in position k on the abacus contributes x k j to the weight of the abacus. The abacus (, v) in the Example 54 has inv(v) = 6, sgn(, v) = ( ) 6 =, and wt(, v) = x 9 x 5 x 0 3x 3 4x 6 5x 0 6 x 4 7x 8 8. Example 57. Let = (3,, ). Table. shows the product of the sign and weight of each abacus with position set µ = + δ(3) = (5, 3, ). Table.: Signs and weights for the set LAbc((3,, )). (, v) LAbc((3,, )) sgn(, v) wt(, v) x 5 x 3 x 3 3 x x 5 x x 5 x x x 3 x x x 3 x 5 x 3 3 x x 3 x 5 3 Summing the monomials from Table. gives the polynomial sgn(, v) wt(, v) = x 5 x 3 x 3 + x 3 x x x x 5 x 3 3 x 3 x 5 x 3 x 5 x x 3 3 x x 3 x 5 3 (,v) LAbc((3,,)) = a (5,3,) (x, x, x 3 ). The next theorem states that this happens in general.

26 Andrew J. Wills Chapter. Background 8 Theorem 58. For all Par N, a +δ(n) (x,..., x N ) = sgn(, v) wt(, v). (,v) LAbc() See [9, p. 46] for a proof. It can be shown [9, p. 458] that a δ(n) (x,..., x N ) = det x N i which is known as the Vandermonde determinant. j i,j N = i<j N (x i x j ), (.).0 Pieri Rules for Schur Polynomials The monomial antisymmetric polynomials provide an alternate algebraic definition of Schur polynomials. Theorem 59. For all Par N, s (x,..., x N ) = a +δ(n)(x,..., x N ) a δ(n) (x,..., x N ). [8, Thm. 3.] and [9, Sec..] give purely bijective proofs of this theorem using abaci, where previous proofs were algebraic (see [, II.5.7] and [9, Thm..45]) or based on the RSK algorithm on tableaux (see [5]). The next theorems, Theorems 60 and 6, are called Pieri rules for Schur polynomials because they describe how to express the product a µ+δ(n) f (or a δ(n) s µ f) in terms of the a s (or a δ(n) s s), where f is h r or e r. The abacus proofs of these Pieri rules and Theorem 59 in [9] all make use of a similar mechanism: beads on an abacus are moved according to certain rules to create a new abacus with a new partition. However, if beads collide while moving, the abacus instead cancels with another abacus with a similar collision. This next theorem describes the product of a monomial antisymmetric polynomial with an elementary symmetric polynomial. Theorem 60. For all µ Par N and all r, a µ+δ(n) (x,..., x N ) e r (x,..., x N ) = Par N : V(µ,r) a +δ(n) (x,..., x N ). See [9, Thm..4] for a bijective proof.

27 Andrew J. Wills Chapter. Background 9 Example 6. Let N = 4 and r =. Then e a (,0 )+δ(4) = a ( 4 )+δ(4) + a (,,0)+δ(4) + a (,0 )+δ(4). Similarly, we can describe the product of a monomial antisymmetric polynomial with a complete symmetric polynomial. Theorem 6. For all µ Par N and all r, a µ+δ(n) (x,..., x N ) h r (x,..., x N ) = Par N : H(µ,r) a +δ(n) (x,..., x N ). See [9, Thm..44] for a bijective proof. Example 63. Let N = 4 and r =. Then h a (,0 )+δ(4) = a (,,0)+δ(4) + a (3,,0 )+δ(4).. Hall-Littlewood Polynomials We now give algebraic definitions for three versions of the Hall-Littlewood polynomials, which are the central objects studied in this work. These definitions can be found in [, Ch. III]. Definition 64. For a partition Par N, variables x,..., x N, and an indeterminate t, define R (x,..., x N ; t) = ( ) w x x x N i tx j N. (.) x w S N i<j i x j We will call R the (unstable) Hall-Littlewood polynomial indexed by. Example 65. For the partition = (,, 0) Par 3, a calculation leads to R (,,0) (x, x, x 3 ) = x x + x x + x x 3 + ( t t )x x x 3 + x x 3 + x x 3 + x x 3. Theorem 66. R is a symmetric polynomial in x,..., x N homogeneous of degree. with coefficients in Z[t] and is See [, III..5] for a proof. For w S N, we can use equation. to write w (x i x j ) = w a δ(n) = sgn(w)a δ(n), i<j

28 Andrew J. Wills Chapter. Background 0 because a δ(n) is antisymmetric. Using this identity, we can rewrite equation. to have the form a δ(n) R (x,..., x N ) = ( ) sgn(w)w x... x N N (x i tx j ). (.3) w S N i<j This expression for a δ(n) R has a number of familiar components that motivate the combinatorial interpretation given in Chapter 3. Theorem 67. For all Par N, the coefficients of R are divisible in Z[t] by i 0 [m i()]! t. See [] for a an algebraic proof. We provide a combinatorial proof in Theorem 0 below. Definition 68. For a partition Par N, variables x,..., x N, and indeterminate t, define P (x,..., x N ; t) = i 0 [m i()]! t R (x,..., x N ). We call P the (stable) Hall-Littlewood polynomial indexed by. Thanks to Theorem 67, the stable Hall-Littlewood polynomial P is also a symmetric polynomial with coefficients in Z[t] and is homogeneous of degree. See Theorem 30 for the motivation for the term stable. Definition 69. For a partition Par N, variables x,..., x N, and indeterminate t, define ( ) Q (x,..., x N ; t) = ( t)( t ) ( t mi() ) P (x,..., x N ; t) (.4) i = ( t) l() i [m i ()]! t P (x,..., x N ; t) = ( t)l() [m 0 ()]! t R (x,..., x N ; t). (.5) We call Q the (variant) Hall-Littlewood polynomial indexed by. Example 70. For the partition = (,, 0) Par 3, R (,,0) = (+t)x x +( t )x x x 3 +( t )x x x 3 +(+t)x x 3+( t )x x x 3+(+t)x x 3, and P (,,0) = x x + ( t)x x x 3 + ( t)x x x 3 + x x 3 + ( t)x x x 3 + x x 3, Q (,,0) = ( t) (+t)x x +( t) ( t )x x x 3 +( t) ( t )x x x 3 +( t) (+t)x x 3 +( t) ( t )x x x 3 + ( t) ( + t)x x 3.

29 Chapter 3 Combinatorial Interpretations of Hall-Littlewood Polynomials We now introduce the objects used to provide a new combinatorial interpretation for a δ(n) R, a δ(n) P, and a δ(n) Q, which will be our focus for the rest of this paper. The objects in question are ordered pairs of abaci and tournaments, called abacus-tournaments. 3. Abacus-Tournaments Definition 7. A tournament τ on vertex set [N] = {,,..., N} is a subset of [N] [N] such that for all i j in [N], exactly one of (i, j) and (j, i) appears in τ, and no pair (i, i) appears in τ. Let T N denote the set of all tournaments with vertex set [N]. Example 7. Let τ = {(, 5), (, 4), (, ), (, 5), (, 3), (4, 5), (4, ), (4, 3), (3, ), (3, 5)}. Then τ is a tournament in T 5. Definition 73. For a partition Par N, an abacus-tournament for is an ordered triple A = (, v, τ) where (, v) is a labeled abacus (see Definition 53) and τ T N is a tournament. Define the word of A to be w(a) = v S N, and the tournament of A to be τ(a) = τ T N. Let AT = {} S N T N denote the set of abacus-tournaments for the partition. An abacus-tournament A = (, v, τ) can be displayed by forming a grid and placing the abacus runner for (, v) on the diagonal of the grid. Each column and row of the grid contains either precisely one bead of the abacus or precisely one bead gap of the abacus. Bead position 0 is at the top left corner. Fill in X s in the grid to encode the ordered pairs in τ: if (v i, v j ) τ, place an X in the column of the bead labeled v i and in the row of the bead labeled v j. Example 74. Let = (, 0 3 ) Par 5 (4), let v = 543 S 5, and let τ T 5 be as in Example 7. Then A = (, v, τ) AT, and A is displayed in Figure 3..

30 Andrew J. Wills Chapter 3. Combinatorial Interpretations of Hall-Littlewood Polynomials Figure 3.: An example of an abacus-tournament Each abacus-tournament A has a signed weight, denoted swt(a), that is calculated from its components via the following definitions. Definition 75. For A = (, v, τ) AT, define the (global) outdegree of a bead v i = k to be out A (k) = {l : (k, l) τ}, the (global) gap count of a bead v i = k to be gap A (k) = i, and upset(a) = {(v j, v i ) τ : i < j}. When the abacus-tournament under consideration is understood from context, we often drop the subscript A, writing out(k) and gap(k). Definition 76. If A = (, v, τ) is an abacus-tournament for, define the signed weight of A to be ( N ) swt(a) = sgn(v)( t) upset(a). (3.) k= x out(k)+gap(k) k Note that the signed weight of an abacus-tournament is a monomial in variables x,..., x N with a coefficient in Z[t]. The signed weight of A can be calculated from the diagram of A as follows. To calculate the exponent of x k in swt(a), add the number of X s in the column of the bead labeled k, namely out(k), to the number of bead gaps on the diagonal northwest of k, namely gap(k). When (v j, v i ) τ with i < j, the corresponding X appears below the main diagonal in the diagram. Consequently, to calculate the signed coefficient of swt(a), raise ( t) to the number of X s below the diagonal in the diagram and multiply by the sign of w(a).

31 Andrew J. Wills Chapter 3. Combinatorial Interpretations of Hall-Littlewood Polynomials 3 Example 77. Let = (, 0 3 ) Par 4 (5) and let A = (, v, τ) be as in Example 74. The bead v = has out() = and gap() =, the bead v = 5 has out(5) = 0 and gap(5) =, the bead v 3 = has out() = 3 and gap() = 0, and so on. Also, there are six X s below the main diagonal corresponding to edges in upset(a) = {(3, ), (3, 5), (4, 5), (4, ), (, ), (, 5)}, so upset(a) = 6. Finally, inv(v) = 4, so swt(a) = ( ) 4 ( t) 6 x 4 x 3 x 3x 3 4x A Combinatorial Interpretation of a δ(n) R We now show that abacus-tournaments for give a combinatorial model for a δ(n) R. Lemma 78. Let Par N and A = (, v, τ) AT. Then swt(a) = sgn(v)x v x N vn x vi (v i,v j ) τ: i<j (v i,v j ) τ: i>j ( tx vi ). Proof. For a given bead v i, edges of the form (v i, v j ) τ where i > j are counted by out(v i ) and upset(a), each such edge contributing a factor of ( t)x vi to swt(a). Edges (v i, v j ) τ such that i < j are only counted by out(v i ), contributing a factor of x vi to swt(a). Therefore ( N ) swt(a) = sgn(v)( t) upset(a) k= x out(k)+gap(k) k ( N = sgn(v)x gap(v ) v x gap(v N ) ( t) upset(a) = sgn(v)x v x N vn v N (v i,v j ) τ: i<j x vi (v i,v j ) τ: i>j k= x out(k) k ) ( tx vi ). (3.) Theorem 79. For a partition Par N, the abacus-tournaments for form a combinatorial model for a δ(n) R (x,..., x N ; t): a δ(n) R = A AT swt(a).

32 Andrew J. Wills Chapter 3. Combinatorial Interpretations of Hall-Littlewood Polynomials 4 Proof. We first show (x i tx j ) = i<j τ T N (i,j) τ: i<j x i (i,j) τ: i>j ( tx i ). In general, to multiply the set of factors of the form (x i tx j ) together, choose either x i or tx j for each factor with i < j and multiply these together. Then add together each of the possible products formed in the previous step. In the first step, each selection of x i or tx j helps to determine a tournament τ T N : if x i is chosen, then (i, j) τ, and if tx j is chosen, then (j, i) τ. The product of the chosen factors encoded by τ is ( tx j ). (i,j) τ: i<j Summing these together and reindexing gives (x i tx j ) = i<j x i τ T N (j,i) τ: i<j (i,j) τ: i<j x i (i,j) τ: i>j ( tx i ). Now, consider the polynomial obtained by summing the signed weights of all abacus-tournaments for a partition Par N. By Lemma 78, swt(a) = sgn(v)x v x N vn x vi ( tx vi ) A AT (,v,τ) AT = (,v) LAbc() = v S N sgn(v)v sgn(v)x v x N vn ( x x N N (v i,v j ) τ :i<j (v i,v j ) τ: i>j τ T N (v i,v j ) τ: i<j x vi ) (x i tx j ). The last line of the above equation matches a δ(n) R by (.3). i<j (v i,v j ) τ :i>j ( tx vi ) Example 80. Table 3. lists the abacus-tournaments for = (, 0) and their signed weights. Observe that A AT swt(a) = x x = a δ() R (x, x ), since by (.3). a δ() R (,0) (x, x ) = x (x tx ) x (x tx ) = x x

33 Andrew J. Wills Chapter 3. Combinatorial Interpretations of Hall-Littlewood Polynomials 5 Table 3.: The signed weights of abacus-tournaments for = (, 0). A AT w(a) τ(a) swt(a) A AT w(a) τ(a) swt(a) {(, )} x {(, )} tx x {(, )} x {(, )} tx x 3.3 Blocks and Involutions Our next goal is to develop combinatorial models for a δ(n) P and a δ(n) Q from the abacustournament model for a δ(n) R. First, we need some technical constructions to help cancel objects in AT. Definition 8. If j and k 0, we call the set of word positions {j, j +,..., j + k} a block. A block B can be visualized on the diagram of an abacus-tournament A by drawing a square box with corners on the main diagonal that captures the beads v j, v j+,..., v j+k where w(a) = v v v N. We call this box the square of block B. Definition 8. Given Par N, an abacus-tournament A = (, v, τ) AT, and a block B = {j, j+,..., j+k}, let the abacus-tournament restricted to block B, denoted A B, refer to the set of beads {v j, v j+,..., v j+k } together with tournament edges {(v i, v l ) τ : i, l B}. Furthermore, define for i B the local outdegree of bead v i in block B to be out B A (v i) = {(v i, v l ) τ : l B}, the local gap count of bead v i in block B to be gap B A (v i) = i j+k, and upset(a B ) = {(v l, v i ) τ : i, l B and i < l}. Visually, the local outdegree of a bead v i can be determined by counting X s in the column of v i that are also contained in the square delimiting the given block. The local gap count

34 Andrew J. Wills Chapter 3. Combinatorial Interpretations of Hall-Littlewood Polynomials 6 Figure 3.: The square of block {,, 3, 4, 5} in an abacus-tournament can be obtained by counting the bead gaps to the left of bead v i that are contained in the square for the given block. If the abacus-tournament A under consideration is understood from context, then we may write out B (v i ) and gap B (v i ), dropping the subscript A. Example 83. The single block B = {,..., N} contains every word position. See Figure 3.. In this and later figures, the portion of an abacus-tournament restricted to a block is outlined in red. Definition 84. Given Par N, define Pos (i) be the set of all positions j for which j = i. Define the collection of -blocks to be B = {Pos (i) : i 0, Pos (i) }. The square for Pos (i) encloses the consecutive beads on any abacus for that have exactly i gaps above them. Note that Pos (i) = m i (), the number of parts of equal to i. Specifically, for i with m i () > 0, Pos (i) contains positions N ( i j=0 m j()) +,..., N ( i j=0 m j()) of v. Likewise, the beads in the (, v, τ) Pos (i) are located in columns ( i j=0 m j()) + i,..., ( i j=0 m j()) + i of the abacus-tournament diagram. Example 85. Let N = 8 and = (3 3, 4, 0). Then Pos (0) = {8}, Pos () = {4, 5, 6, 7} and Pos (3) = {,, 3}. All other -blocks for this are empty. Figure 3.3 colors the squares of the -blocks in an abacus-tournament for. Definition 86. A set of blocks B is non-overlapping iff there does not exist a bead position that is contained in more than one block of B. For example, the set of -blocks in Example 85 is non-overlapping. The /µ-blocks from Definition 39 and Example 40 below give a more complicated example of non-overlapping blocks. Definition 87. Given Par N, we say that two -blocks Pos (i) and Pos (j) with i < j are adjacent if Pos (k) = for all i < k < j. In general, if B = {b,, b s }, C = {c,..., c r } are blocks, we say B and C are adjacent for iff b = c r + and cr b > 0.

35 Andrew J. Wills Chapter 3. Combinatorial Interpretations of Hall-Littlewood Polynomials 7 Figure 3.3: The squares of the -blocks in an abacus-tournament Example 88. In Figure 3.3, Pos (0) and Pos () are adjacent and Pos () and Pos (3) are adjacent. There are no other pairs of adjacent -blocks in Figure Global and Local Exponent Collisions Definition 89. An abacus-tournament A = (, v, τ) is said to have a (global) exponent collision if there exist l, m such that out(v l ) + gap(v l ) = out(v m ) + gap(v m ). If B is a block, then we say A has a local exponent collision in B if there exist l, m B such that out B (v l ) + gap B (v l ) = out B (v m ) + gap B (v m ). Example 90. Let N = 5 and = (, 0 3 ). The abacus-tournament in Figure 3. has a (global) exponent collision between beads v = 5 and v 5 = 3 because out(5) + gap(5) = 0 + = + 0 = out(3) + gap(3). Example 9. Let N = 8 and = (3 3, 4, 0). The abacus-tournament in Figure 3.3 has a local exponent collision in block B = Pos () between beads v 5 = 6 and v 7 = 3 because out B (6) + gap B (6) = + 0 = out B (3) + gap B (3).

36 Andrew J. Wills Chapter 3. Combinatorial Interpretations of Hall-Littlewood Polynomials 8 Definition 9. If is a partition and B is a set of non-overlapping blocks, let AT B denote the set of abacus-tournaments in AT with no local exponent collisions in any of the blocks of B. Given a partition and a set of non-overlapping blocks B, we can pair together abacustournaments with local exponent collisions in B in such a way that the two abacus-tournaments in each pair have the same weight but opposing signs. As pairs, matched abacus-tournaments with local exponent collisions then contribute a combined signed weight of zero to the total sum of signed weights. With this pairing, the set of all abacus-tournaments with local exponent collisions make no net contribution to the sum of signed weights. So, a sum over only abacus-tournaments without local exponent collisions gives the same sum of signed weights as a sum over all abacus-tournaments. Consider the following examples to see how such a pairing can work. Example 93. Let N = 5, let = (, 0 3 ), let B = {,, 3, 4, 5}, and let A = (, v, τ) be the abacus-tournament in Figure 3.. Recall from Example 74 that the abacus (, v) has position set pos(a) = + δ(n) = (6, 5,,, 0) and w(a) = v = 543. The tournament τ is given by τ = {(, 5), (, 4), (, 3), (, 5), (, ), (4, 3), (4, ), (4, 5), (3, 5), (3, )}. The abacustournament A has signed weight ( ) 4 ( t) 6 x 4 x 3 x 3x 3 4x 5. Since the block B includes every position, local outdegree and gap counts relative to B will be equal to its global outdegree and gap counts of A. The abacus-tournament A has exponent collisions: for example, x 3 and x 5 have equal exponents in swt(a). Consider what happens to the abacus-tournament A if the bead labeled 3 in position 5 and the bead labeled 5 in position switch positions, and the X s in the abacus-tournament diagram do not move locations to reflect this change. This produces the new abacus-tournament A = (, v, τ ) in Figure 3.4, where pos(a ) = + δ(n), w(a ) = v = 345, and τ = {(, 4), (, 3), (, 5), (, 3), (, ), (4, 5), (4, ), (4, 3), (5, 3), (5, )}. In the new tournament τ, every instance of a 3 in an ordered pair in τ has been replaced by a 5 and vice versa. The new abacus-tournament has signed weight ( ) ( t) 6 x 4 x 3 x 3x 3 4x 5, so the sum of swt(a) and swt(a ) is zero. Notice that A also has an exponent collision between the bead labeled 3, now in position, and the bead labeled 5, now in position 5. Switching these beads without moving any X s in the diagram of A restores the original abacus-tournament A. Example 94. Let N = 8 and = (3 3, 4, 0), and consider the set of -blocks B and the abacus-tournament A = (, v, τ) in Figure 3.3. A has signed weight ( ) 8 ( t) 8 x x 9 x 4 3x 7 4x 5x 6 6x 4 7x 8 8. The previous example suggests switching beads labeled 3 and 7 because x 3 and x 7 have matching exponent values of 4 in swt(a). However, for the choice of blocks in our current