Counting Permutations in $S_{2n}$ and $S_{2n+1}$
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866914864692199424 |
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| author | Luo, Yuewen |
| author_facet | Luo, Yuewen |
| contents | Let $α(n)$ denote the number of perfect square permutations in the symmetric group $S_n$. The conjecture $α(2n+1) = (2n+1) α(2n)$, provided by Stanley[4], was proved by Blum[1] using a generating function. This paper presents a combinatorial proof for this conjecture. At the same time, we demonstrate that all permutations with an even number of even cycles in both $S_{2n}$ and $S_{2n+1}$ can be categorized into three distinct types that correspond to each other. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_07366 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Counting Permutations in $S_{2n}$ and $S_{2n+1}$ Luo, Yuewen Combinatorics Let $α(n)$ denote the number of perfect square permutations in the symmetric group $S_n$. The conjecture $α(2n+1) = (2n+1) α(2n)$, provided by Stanley[4], was proved by Blum[1] using a generating function. This paper presents a combinatorial proof for this conjecture. At the same time, we demonstrate that all permutations with an even number of even cycles in both $S_{2n}$ and $S_{2n+1}$ can be categorized into three distinct types that correspond to each other. |
| title | Counting Permutations in $S_{2n}$ and $S_{2n+1}$ |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2407.07366 |