Polynomial Expressions for the Dimensions of the Representations of Symmetric Groups and Restricted Standard Young Tableaux
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914983558774784 |
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| author | Cohen, Avichai Zemel, Shaul |
| author_facet | Cohen, Avichai Zemel, Shaul |
| contents | Given a partition $λ$ of a number $k$, it is known that by adding a long line of length $n-k$, the dimension of the associated representation of $S_{n}$ is an integer-valued polynomial of degree $k$ in $n$. We show that its expansion in the binomial basis is bounded by the length of $λ$, and that the resulting coefficient of index $h$, with alternating signs, counts the standard Young tableaux of shape $λ$ in which a given collection of consecutive $h$ numbers lie in increasing rows. We also construct bijections in order to demonstare explicitly that this number is indeed independent of the set of consecutive $h$ numbers used. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_16758 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Polynomial Expressions for the Dimensions of the Representations of Symmetric Groups and Restricted Standard Young Tableaux Cohen, Avichai Zemel, Shaul Combinatorics 05A17, 05A19, 20C30, 05E10 Given a partition $λ$ of a number $k$, it is known that by adding a long line of length $n-k$, the dimension of the associated representation of $S_{n}$ is an integer-valued polynomial of degree $k$ in $n$. We show that its expansion in the binomial basis is bounded by the length of $λ$, and that the resulting coefficient of index $h$, with alternating signs, counts the standard Young tableaux of shape $λ$ in which a given collection of consecutive $h$ numbers lie in increasing rows. We also construct bijections in order to demonstare explicitly that this number is indeed independent of the set of consecutive $h$ numbers used. |
| title | Polynomial Expressions for the Dimensions of the Representations of Symmetric Groups and Restricted Standard Young Tableaux |
| topic | Combinatorics 05A17, 05A19, 20C30, 05E10 |
| url | https://arxiv.org/abs/2410.16758 |