An Extension of Pólya's Enumeration Theorem
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866915148919209984 |
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| author | Zhan, Xiongfeng Huang, Xueyi |
| author_facet | Zhan, Xiongfeng Huang, Xueyi |
| contents | In combinatorics, Pólya's Enumeration Theorem is a powerful tool for solving a wide range of counting problems, including the enumeration of groups, graphs, and chemical compounds. In this paper, we present an extension of Pólya's Enumeration Theorem. As an application, we derive a formula that expresses the $n$-th elementary symmetric polynomial in $m$ indeterminates (where $n\leq m$) as a variant of the cycle index polynomial of the symmetric group $\mathrm{Sym}(n)$. This result resolves a problem posed by Amdeberhan in 2012. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_12508 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An Extension of Pólya's Enumeration Theorem Zhan, Xiongfeng Huang, Xueyi Combinatorics 05A19 In combinatorics, Pólya's Enumeration Theorem is a powerful tool for solving a wide range of counting problems, including the enumeration of groups, graphs, and chemical compounds. In this paper, we present an extension of Pólya's Enumeration Theorem. As an application, we derive a formula that expresses the $n$-th elementary symmetric polynomial in $m$ indeterminates (where $n\leq m$) as a variant of the cycle index polynomial of the symmetric group $\mathrm{Sym}(n)$. This result resolves a problem posed by Amdeberhan in 2012. |
| title | An Extension of Pólya's Enumeration Theorem |
| topic | Combinatorics 05A19 |
| url | https://arxiv.org/abs/2412.12508 |