On the Limiting Vacillating Tableaux for Integer Sequences
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arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866912163993485312 |
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| author | Berikkyzy, Zhanar Harris, Pamela E. Pun, Anna Yan, Catherine Zhao, Chenchen |
| author_facet | Berikkyzy, Zhanar Harris, Pamela E. Pun, Anna Yan, Catherine Zhao, Chenchen |
| contents | A fundamental identity in the representation theory of the partition algeba is $n^k = \sum_λ f^λm_k^λ$ for $n \geq 2k$, where $λ$ ranges over integer partitions of $n$, $f^λ$ is the number of standard Young tableaux of shape $λ$, and $m_k^λ$ is the number of vacillating tableaux of shape $λ$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_13091 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On the Limiting Vacillating Tableaux for Integer Sequences Berikkyzy, Zhanar Harris, Pamela E. Pun, Anna Yan, Catherine Zhao, Chenchen Combinatorics 05A15, 05A18, 05E10 A fundamental identity in the representation theory of the partition algeba is $n^k = \sum_λ f^λm_k^λ$ for $n \geq 2k$, where $λ$ ranges over integer partitions of $n$, $f^λ$ is the number of standard Young tableaux of shape $λ$, and $m_k^λ$ is the number of vacillating tableaux of shape $λ$ 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. |
| title | On the Limiting Vacillating Tableaux for Integer Sequences |
| topic | Combinatorics 05A15, 05A18, 05E10 |
| url | https://arxiv.org/abs/2208.13091 |