scholarly journals Two Remarks on Skew Tableaux

10.37236/2012 ◽  
2011 ◽  
Vol 18 (2) ◽  
Author(s):  
Richard P. Stanley
Keyword(s):  
Generating Functions ◽  
Schur Function ◽  
Young Tableaux ◽  
Skew Schur Function

This paper contains two results on the number $f^{\sigma/\tau}$ of standard skew Young tableaux of shape $\sigma/\tau$. The first concerns generating functions for certain classes of "periodic" shapes related to work of Gessel-Viennot and Baryshnikov-Romik. The second result gives an evaluation of the skew Schur function $s_{\lambda/\mu}(x)$ at $x=(1,1/2^{2k},1/3^{2k}, \dots)$ for $k=1,2,3$ in terms of $f^{\sigma/\tau}$ for a certain skew shape $\sigma/\tau$ depending on $\lambda/\mu$.

scholarly journals Lattice Path Proofs of Extended Bressoud-Wei and Koike Skew Schur Function Identities

10.37236/534 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
A. M. Hamel ◽  
R. C. King
Keyword(s):  
Lattice Path ◽  
Schur Function ◽  
Skew Schur Function

A recent paper of the present authors provides extensions to two classical determinantal results of Bressoud and Wei, and of Koike. The proofs in that paper were algebraic. The present paper contains combinatorial lattice path proofs.

scholarly journals Crystals and Schur $P$-Positive Expansions

10.37236/7557 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
Seung-Il Choi ◽  
Jae-Hoon Kwon
Keyword(s):  
Crystal Structures ◽  
Schur Function ◽  
Skew Schur Function

We give a new characterization of Littlewood-Richardson-Stembridge tableaux for Schur $P$-functions by using the theory of $\mathfrak{q}(n)$-crystals. We also give alternate proofs of the Schur $P$-expansion of a skew Schur function due to Ardila and Serrano, and the Schur expansion of a Schur $P$-function due to Stembridge using the associated crystal structures.

On the product of a Schur function and a skew Schur function

1993 ◽  
Vol 34 (2) ◽  
pp. 305-315
Author(s):  
Toshihiro Watanabe
Keyword(s):  
Schur Function ◽  
Skew Schur Function

scholarly journals Extended Bressoud–Wei and Koike skew Schur function identities

2011 ◽  
Vol 118 (2) ◽  
pp. 545-557
Author(s):  
A.M. Hamel ◽  
R.C. King
Keyword(s):  
Schur Function ◽  
Skew Schur Function

scholarly journals Generating Functions for Inverted Semistandard Young Tableaux and Generalized Ballot Numbers

10.37236/6376 ◽  
2017 ◽  
Vol 24 (2) ◽  
Author(s):  
Paul Drube
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Young Tableau ◽  
Young Tableaux ◽  
Dyck Paths ◽  
Standard Tableau ◽  
Ballot Numbers ◽  
Standard Young Tableaux

An inverted semistandard Young tableau is a row-standard tableau along with a collection of inversion pairs that quantify how far the tableau is from being column semistandard. Such a tableau with precisely $k$ inversion pairs is said to be a $k$-inverted semistandard Young tableau. Building upon earlier work by Fresse and the author, this paper develops generating functions for the numbers of $k$-inverted semistandard Young tableaux of various shapes $\lambda$ and contents $\mu$. An easily-calculable generating function is given for the number of $k$-inverted semistandard Young tableaux that "standardize" to a fixed semistandard Young tableau. For $m$-row shapes $\lambda$ and standard content $\mu$, the total number of $k$-inverted standard Young tableaux of shape $\lambda$ is then enumerated by relating such tableaux to $m$-dimensional generalizations of Dyck paths and counting the numbers of "returns to ground" in those paths. In the rectangular specialization of $\lambda = n^m$ this yields a generating function that involves $m$-dimensional analogues of the famed Ballot numbers. Our various results are then used to directly enumerate all $k$-inverted semistandard Young tableaux with arbitrary content and two-row shape $\lambda = a^1 b^1$, as well as all $k$-inverted standard Young tableaux with two-column shape $\lambda=2^n$.

scholarly journals Stable Equivalence over Symmetric Functions

10.37236/1880 ◽  
2005 ◽  
Vol 11 (2) ◽  
Author(s):  
William Y. C. Chen ◽  
Arthur L. B. Yang
Keyword(s):  
Symmetric Functions ◽  
Affirmative Answer ◽  
Schur Function ◽  
Stable Equivalence ◽  
Skew Schur Function

By using cutting strips and transformations on outside decompositions of a skew diagram, we show that the Giambelli-type matrices for a given skew Schur function are stably equivalent to each other over symmetric functions. As a consequence, the Jacobi-Trudi matrix and the transpose of the dual Jacobi-Trudi matrix are stably equivalent over symmetric functions. This leads to an affirmative answer to a question proposed by Kuperberg.

scholarly journals A Macdonald Vertex Operator and Standard Tableaux Statistics

10.37236/1383 ◽  
1998 ◽  
Vol 5 (1) ◽  
Author(s):  
Mike Zabrocki
Keyword(s):  
Vertex Operator ◽  
Generating Functions ◽  
Symmetric Function ◽  
Symmetric Functions ◽  
Schur Function ◽  
Inductive Proof ◽  
Integer Coefficients ◽  
The Family ◽  
The Relationship ◽  
Two Parameter

The two parameter family of coefficients $K_{\lambda \mu}(q,t)$ introduced by Macdonald are conjectured to $(q,t)$ count the standard tableaux of shape $\lambda $. If this conjecture is correct, then there exist statistics $a_\mu(T)$ and $b_\mu(T)$ such that the family of symmetric functions $H_\mu[X;q,t] = \sum_\lambda K_{\lambda \mu}(q,t) s_\lambda [X]$ are generating functions for the standard tableaux of size $|\mu|$ in the sense that $H_\mu[X;q,t] = \sum_{T} q^{a_\mu(T)} t^{b_\mu(T)} s_{\lambda (T)}[X]$ where the sum is over standard tableau of of size $|\mu|$. We give a formula for a symmetric function operator $H_2^{qt}$ with the property that $H_2^{qt} H_{(2^a1^b)}[X;q,t]= H_{(2^{a+1}1^b)}[X;q,t]$. This operator has a combinatorial action on the Schur function basis. We use this Schur function action to show by induction that $H_{(2^a1^b)}[X;q,t]$ is the generating function for standard tableaux of size $2a+b$ (and hence that $K_{\lambda (2^a1^b)}(q,t)$ is a polynomial with non-negative integer coefficients). The inductive proof gives an algorithm for 'building' the standard tableaux of size $n+2$ from the standard tableaux of size $n$ and divides the standard tableaux into classes that are generalizations of the catabolism type. We show that reversing this construction gives the statistics $a_\mu(T)$ and $b_\mu(T)$ when $\mu$ is of the form $(2^a1^b)$ and that these statistics prove conjectures about the relationship between adjacent rows of the $(q,t)$-Kostka matrix that were suggested by Lynne Butler.

scholarly journals Families of Major Index Distributions: Closed Forms and Unimodality

10.37236/8585 ◽  
2019 ◽  
Vol 26 (3) ◽  
Author(s):  
William J. Keith
Keyword(s):  
Schur Functions ◽  
Positive Answer ◽  
Schur Function ◽  
Young Tableaux ◽  
Large Collection ◽  
Major Index ◽  
The Family ◽  
Standard Young Tableaux ◽  
Principal Specialization

Closed forms for $f_{\lambda,i} (q) := \sum_{\tau \in SYT(\lambda) : des(\tau) = i} q^{maj(\tau)}$, the distribution of the major index over standard Young tableaux of given shapes and specified number of descents, are established for a large collection of $\lambda$ and $i$. Of particular interest is the family that gives a positive answer to a question of Sagan and collaborators. All formulas established in the paper are unimodal, most by a result of Kirillov and Reshetikhin. Many can be identified as specializations of Schur functions via the Jacobi-Trudi identities. If the number of arguments is sufficiently large, it is shown that any finite principal specialization of any Schur function $s_\lambda(1,q,q^2,\dots,q^{n-1})$ has a combinatorial realization as the distribution of the major index over a given set of tableaux.

scholarly journals Hook-content Formulae for Symplectic and Orthogonal Tableaux

2012 ◽  
Vol 55 (3) ◽  
pp. 462-473
Author(s):  
Peter S. Campbell ◽  
Anna Stokke
Keyword(s):  
Young Diagram ◽  
Schur Functions ◽  
Schur Function ◽  
Young Tableaux

AbstractBy considering the specialisation sλ(1, q, q2, … , qn–1) of the Schur function, Stanley was able to describe a formula for the number of semistandard Young tableaux of shape λ in terms of the contents and hook lengths of the boxes in the Young diagram. Using specialisations of symplectic and orthogonal Schur functions, we derive corresponding formulae, first given by El Samra and King, for the number of semistandard symplectic and orthogonal λ-tableaux.

scholarly journals Cyclic Sieving and Plethysm Coefficients

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
David B Rush
Keyword(s):  
Fixed Points ◽  
Schur Function ◽  
Young Tableaux ◽  
Jeu De Taquin ◽  
Cyclic Action ◽  
Domino Tableaux ◽  
Ribbon Tableaux ◽  
International Audience ◽  
Cyclic Sieving ◽  
Element Set

International audience A combinatorial expression for the coefficient of the Schur function $s_{\lambda}$ in the expansion of the plethysm $p_{n/d}^d \circ s_{\mu}$ is given for all $d$ dividing $n$ for the cases in which $n=2$ or $\lambda$ is rectangular. In these cases, the coefficient $\langle p_{n/d}^d \circ s_{\mu}, s_{\lambda} \rangle$ is shown to count, up to sign, the number of fixed points of an $\langle s_{\mu}^n, s_{\lambda} \rangle$-element set under the $d^e$ power of an order $n$ cyclic action. If $n=2$, the action is the Schützenberger involution on semistandard Young tableaux (also known as evacuation), and, if $\lambda$ is rectangular, the action is a certain power of Schützenberger and Shimozono's <i>jeu-de-taquin</i> promotion.This work extends results of Stembridge and Rhoades linking fixed points of the Schützenberger actions to ribbon tableaux enumeration. The conclusion for the case $n=2$ is equivalent to the domino tableaux rule of Carré and Leclerc for discriminating between the symmetric and antisymmetric parts of the square of a Schur function. Une expression combinatoire pour le coefficient de la fonction de Schur $s_{\lambda}$ dans l’expansion du pléthysme $p_{n/d}^d \circ s_{\mu}$ est donné pour tous $d$ que disent $n$, dans les cas où $n=2$, ou $\lambda$ est rectangulaire. Dans ces cas, le coefficient $\langle p_{n/d}^d \circ s_{\mu}, s_{\lambda} \rangle$ se montre à compter, où l’on ignore le signe, le nombre des point fixés d’un ensemble de $\langle s_{\mu}^n, s_{\lambda} \rangle$ éléments sous la puissance $d^e$ d’une action cyclique de l’ordre $n$. Si $n=2$, l’action est l’involution de Schützenberger sur les tableaux semi-standard de Young (aussi connu sous le nom des évacuations), et si $\lambda$ est rectangulaire, l’action est une certaine puissance de l’avancement jeu-de-taquin de Schützenberger et Shimozono.Ce travail étend les résultats de Stembridge et Rhoades, liant les point fixés des actions de Schützenberger aux tableaux de ruban. Pour le cas $n=2$ , la conclusion est équivalent à la règle des tableaux de dominos de Carré et Leclerc, qui distingue entre les parties symétriques et asymétriques du carré d’une fonction de Schur.

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