A bijection for tuples of commuting permutations and a log-concavity conjecture
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866911841302609920 |
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| author | Abdesselam, Abdelmalek Brunialti, Pedro Doan, Tristan Velie, Philip |
| author_facet | Abdesselam, Abdelmalek Brunialti, Pedro Doan, Tristan Velie, Philip |
| contents | Let $A(\ell,n,k)$ denote the number of $\ell$-tuples of commuting permutations of $n$ elements whose permutation action results in exactly $k$ orbits or connected components. We provide a new proof of an explicit formula for $A(\ell,n,k)$ which is essentially due to Bryan and Fulman, in their work on orbifold higher equivariant Euler characteristics. Our proof is self-contained, elementary, and relies on the construction of an explicit bijection, in order to perform the $\ell+1\rightarrow \ell$ reduction. We also investigate a conjecture by the first author, regarding the log-concavity of $A(\ell,n,k)$ with respect to $k$. The conjecture generalizes a previous one by Heim and Neuhauser related to the Nekrasov-Okounkov formula. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_09407 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A bijection for tuples of commuting permutations and a log-concavity conjecture Abdesselam, Abdelmalek Brunialti, Pedro Doan, Tristan Velie, Philip Combinatorics Number Theory Probability Let $A(\ell,n,k)$ denote the number of $\ell$-tuples of commuting permutations of $n$ elements whose permutation action results in exactly $k$ orbits or connected components. We provide a new proof of an explicit formula for $A(\ell,n,k)$ which is essentially due to Bryan and Fulman, in their work on orbifold higher equivariant Euler characteristics. Our proof is self-contained, elementary, and relies on the construction of an explicit bijection, in order to perform the $\ell+1\rightarrow \ell$ reduction. We also investigate a conjecture by the first author, regarding the log-concavity of $A(\ell,n,k)$ with respect to $k$. The conjecture generalizes a previous one by Heim and Neuhauser related to the Nekrasov-Okounkov formula. |
| title | A bijection for tuples of commuting permutations and a log-concavity conjecture |
| topic | Combinatorics Number Theory Probability |
| url | https://arxiv.org/abs/2309.09407 |