On the Correspondence Between Integer Sequences and Vacillating Tableaux

TL;DR

This paper explores the detailed properties of a bijection linking integer sequences to vacillating tableaux, extending understanding of their correspondence in the context of partition algebra representation theory.

Contribution

It characterizes the integer sequences that map to shapes with specific first parts under the bijection, broadening the theoretical understanding of this combinatorial correspondence.

Findings

Characterization of sequences with shape first part equal to n

Analysis of sequences with shape first part equal to n-k

Extension of the bijection to general n and k

Abstract

A fundamental identity in the representation theory of the partition algebra is $n^k = \sum_{\lambda} f^\lambda m_k^\lambda$ for $n \geq 2k$, where $\lambda$ ranges over integer partitions of $n$, $f^\lambda$ is the number of standard Young tableaux of shape $\lambda$, and $m_k^\lambda$ is the number of vacillating tableaux of shape $\lambda$ and length $2k$. Using a combination of RSK insertion and jeu de taquin, Halverson and Lewandowski constructed a bijection $DI_n^k$ that maps each integer sequence in $[n]^k$ to a pair of tableaux of the same shape, where one is a standard Young tableau and the other is a vacillating tableau. In this paper, we study the fine properties of Halverson and Lewandowski's bijection and explore the correspondence between integer sequences and the vacillating tableaux via the map $DI_n^k$ for general integers $n$ and $k$. In particular, we characterize the integer sequences $\boldsymbol{i}$ whose corresponding shape, $\lambda$, in the image $DI_n^k(\boldsymbol{i})$, satisfies $\lambda_1 = n$ or $\lambda_1 = n-k$.