Log-concavity with respect to the number of orbits for infinite tuples of commuting permutations
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866929206291595264 |
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| author | Abdesselam, Abdelmalek |
| author_facet | Abdesselam, Abdelmalek |
| contents | Let $A(p,n,k)$ be the number of $p$-tuples of commuting permutations of $n$ elements whose permutation action results in exactly $k$ orbits or connected components. We formulate the conjecture that, for every fixed $p$ and $n$, the $A(p,n,k)$ form a log-concave sequence with respect to $k$. For $p=1$ this is a well known property of unsigned Stirling numbers of the first kind. As the $p=2$ case, our conjecture includes a previous one by Heim and Neuhauser, which strengthens a unimodality conjecture for the Nekrasov-Okounkov hook length polynomials. In this article, we prove the $p=\infty$ case of our conjecture. We start from an expression for the $A(p,n,k)$ which follows from an identity by Bryan and Fulman, obtained in the their study of orbifold higher equivariant Euler characteristics. We then derive the $p\rightarrow\infty$ asymptotics. The last step essentially amounts to the log-concavity in $k$ of a generalized Turán number, namely, the maximum product of $k$ positive integers whose sum is $n$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2309_07358 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Log-concavity with respect to the number of orbits for infinite tuples of commuting permutations Abdesselam, Abdelmalek Combinatorics Number Theory Probability Let $A(p,n,k)$ be the number of $p$-tuples of commuting permutations of $n$ elements whose permutation action results in exactly $k$ orbits or connected components. We formulate the conjecture that, for every fixed $p$ and $n$, the $A(p,n,k)$ form a log-concave sequence with respect to $k$. For $p=1$ this is a well known property of unsigned Stirling numbers of the first kind. As the $p=2$ case, our conjecture includes a previous one by Heim and Neuhauser, which strengthens a unimodality conjecture for the Nekrasov-Okounkov hook length polynomials. In this article, we prove the $p=\infty$ case of our conjecture. We start from an expression for the $A(p,n,k)$ which follows from an identity by Bryan and Fulman, obtained in the their study of orbifold higher equivariant Euler characteristics. We then derive the $p\rightarrow\infty$ asymptotics. The last step essentially amounts to the log-concavity in $k$ of a generalized Turán number, namely, the maximum product of $k$ positive integers whose sum is $n$. |
| title | Log-concavity with respect to the number of orbits for infinite tuples of commuting permutations |
| topic | Combinatorics Number Theory Probability |
| url | https://arxiv.org/abs/2309.07358 |