On log-concavity of the number of orbits in commuting tuples of permutations
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| Format: | Preprint |
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2024
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| _version_ | 1866916375664001024 |
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| author | Tripathi, Raghavendra |
| author_facet | Tripathi, Raghavendra |
| contents | Denote by $A(p, n, k)$ the number of commuting $p$-tuples of permutations on $[n]$ that have exactly $k$ distinct orbits. It was conjectured in~\cite{abdesselam2023log} that $A(p, n, k)$ is log-concave with respect to $k$ for every $p\geq 2, n\geq 3$, and the log-concavity was proved in ``$p=\infty$" case. In this paper, we prove that for $k=n-α$, the log-concavity for $A(p, n, k)$ holds for every $p\geq 2$ for sufficiently large $n$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_17212 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On log-concavity of the number of orbits in commuting tuples of permutations Tripathi, Raghavendra Combinatorics Number Theory 05A17 Denote by $A(p, n, k)$ the number of commuting $p$-tuples of permutations on $[n]$ that have exactly $k$ distinct orbits. It was conjectured in~\cite{abdesselam2023log} that $A(p, n, k)$ is log-concave with respect to $k$ for every $p\geq 2, n\geq 3$, and the log-concavity was proved in ``$p=\infty$" case. In this paper, we prove that for $k=n-α$, the log-concavity for $A(p, n, k)$ holds for every $p\geq 2$ for sufficiently large $n$. |
| title | On log-concavity of the number of orbits in commuting tuples of permutations |
| topic | Combinatorics Number Theory 05A17 |
| url | https://arxiv.org/abs/2408.17212 |