On the Number of Partition Weights with Kostka Multiplicity One

Given a positive integer $n$, and partitions $\lambda$ and $\mu$ of $n$, let $K_{\lambda \mu}$ denote the Kostka number, which is the number of semistandard Young tableaux of shape $\lambda$ and weight $\mu$. Let $J(\lambda)$ denote the number of $\mu$ such that $K_{\lambda \mu} = 1$. By applying a result of Berenshtein and Zelevinskii, we obtain a formula for $J(\lambda)$ in terms of restricted partition functions, which is recursive in the number of distinct part sizes of $\lambda$. We use this to classify all partitions $\lambda$ such that $J(\lambda) = 1$ and all $\lambda$ such that $J(\lambda) = 2$. We then consider signed tableaux, where a semistandard signed tableau of shape $\lambda$ has entries from the ordered set $\{0 < \bar{1} < 1 < \bar{2} < 2 < \cdots \}$, and such that $i$ and $\bar{i}$ contribute equally to the weight. For a weight $(w_0, \mu)$ with $\mu$ a partition, the signed Kostka number $K^{\pm}_{\lambda,(w_0, \mu)}$ is defined as the number of semistandard signed tableaux of shape $\lambda$ and weight $(w_0, \mu)$, and $J^{\pm}(\lambda)$ is then defined to be the number of weights $(w_0, \mu)$ such that $K^{\pm}_{\lambda, (w_0, \mu)} = 1$. Using different methods than in the unsigned case, we find that the only nonzero value which $J^{\pm}(\lambda)$ can take is $1$, and we find all sequences of partitions with this property. We conclude with an application of these results on signed tableaux to the character theory of finite unitary groups.

Partitions, Kostka Polynomials and Pairs of Trees

Bennett et al. presented a recursive algorithm to create a family of partitions from one or several partitions. They were mainly interested in the cases when we begin with a single square partition or with several partitions with only one part. The cardinalities of those families of partitions are the Catalan and ballot numbers, respectively. In this paper we present a non-recursive description for those families and prove that the generating function of the size of those partitions is a Kostka number. We also present bijections between those sets of partitions and sets of trees and forests enumerated by the Catalan an ballot numbers, respectively.

A Relationship between the Major Index for Tableaux and the Charge Statistic for Permutations

The widely studied $q$-polynomial $f^{\lambda}(q)$, which specializes when $q=1$ to $f^{\lambda}$, the number of standard Young tableaux of shape $\lambda$, has multiple combinatorial interpretations. It represents the dimension of the unipotent representation $S_q^{\lambda}$ of the finite general linear group $GL_n(q)$, it occurs as a special case of the Kostka-Foulkes polynomials, and it gives the generating function for the major index statistic on standard Young tableaux. Similarly, the $q$-polynomial $g^{\lambda}(q)$ has combinatorial interpretations as the $q$-multinomial coefficient, as the dimension of the permutation representation $M_q^{\lambda}$ of the general linear group $GL_n(q)$, and as the generating function for both the inversion statistic and the charge statistic on permutations in $W_{\lambda}$. It is a well known result that for $\lambda$ a partition of $n$, $dim(M_q^{\lambda}) = \Sigma_{\mu} K_{\mu \lambda} dim(S_q^{\mu})$, where the sum is over all partitions $\mu$ of $n$ and where the Kostka number $K_{\mu \lambda}$ gives the number of semistandard Young tableaux of shape $\mu$ and content $\lambda$. Thus $g^{\lambda}(q) - f^{\lambda}(q)$ is a $q$-polynomial with nonnegative coefficients. This paper gives a combinatorial proof of this result by defining an injection $f$ from the set of standard Young tableaux of shape $\lambda$, $SYT(\lambda)$, to $W_{\lambda}$ such that $maj(T) = ch(f(T))$ for $T \in SYT(\lambda)$.