On the Limiting Vacillating Tableaux for Integer Sequences

A fundamental identity in the representation theory of the partition algeba is $n^k={\sum }_{\lambda }{f}^{\lambda }{m}_k^{\lambda }$ for $n\ge 2k$, 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 $D{I}_n^k$ 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 $\mathit{i}$, when $n$ is sufficiently large, the vacillating tableaux determined by $D{I}_n^k(\mathit{i})$ become stable when $n\to \mathrm{\infty }$; the limit is called the limiting vacillating tableau for $\mathit{i}$. We give a characterization of the set of limiting vacillating tableaux and presents explicit formulas that enumerate those vacillating tableaux.