标题： 整数序列与摆动表之间的对应关系 显示英文标题

标题： On the Correspondence Between Integer Sequences and Vacillating Tableaux

摘要： 一个在分划代数表示理论中的基本恒等式是$n^k = \sum_{\lambda} f^\lambda m_k^\lambda$对于$n \geq 2k$，其中$\lambda$范围是整数分划的$n$，$f^\lambda$是形状为$\lambda$的标准杨表的数量，而$m_k^\lambda$是形状为$\lambda$且长度为$2k$的振荡表的数量。 使用RSK插入和 jeu de taquin，Halverson和Lewandowski构造了一个双射$DI_n^k$，该双射将$[n]^k$中的每个整数序列映射到一对形状相同的表格，其中一个是标准杨表，另一个是摆动表格。 在本文中，我们研究Halverson和Lewandowski双射的精细性质，并通过映射$DI_n^k$探讨一般整数$n$和$k$之间的整数序列与摆动表格之间的对应关系。 特别是，我们表征整数序列$\boldsymbol{i}$，其在图像$DI_n^k(\boldsymbol{i})$中对应的形状$\lambda$满足$\lambda_1 = n$或$\lambda_1 = n-k$。 显示英文摘要

摘要： 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 $\lambda_1 = n-k$.