On the Limiting Vacillating Tableaux for Integer Sequences

TL;DR

This paper studies the behavior of vacillating tableaux associated with integer sequences as the sequence length grows, establishing stability and explicit enumeration formulas for their limiting forms.

Contribution

It introduces the concept of limiting vacillating tableaux for integer sequences and characterizes their structure and enumeration as the sequence length tends to infinity.

Findings

Vacillating tableaux stabilize for large n

Explicit formulas for counting limiting vacillating tableaux

Characterization of the set of limiting vacillating tableaux

Abstract

A fundamental identity in the representation theory of the partition algeba 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 consisting of a standard Young tableau and a vacillating tableau. In this paper, we show that for a given integer sequence $\boldsymbol{i}$, when $n$ is sufficiently large, the vacillating tableaux determined by $DI_n^k(\boldsymbol{i})$ become stable when $n \rightarrow \infty$; the limit is called the limiting vacillating tableau for $\boldsymbol{i}$. We give a characterization of the set of limiting vacillating tableaux and presents explicit formulas that enumerate those vacillating tableaux.