Asymptotics of commuting $\ell$-tuples in symmetric groups and log-concavity

Kathrin Bringmann; Johann Franke; Bernhard Heim

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arXiv · 2401.05874

Asymptotics of commuting $\ell$-tuples in symmetric groups and log-concavity

Kathrin Bringmann·Johann Franke·Bernhard Heim

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Abstract

Denote by $N_{\ell}(n)$ the number of $\ell$-tuples of elements in the symmetric group $S_n$ with commuting components, normalized by the order of $S_n$. In this paper, we prove asymptotic formulas for $N_\ell(n)$. In addition, general criteria for log-concavity are shown, which can be applied to $N_\ell(n)$ among other examples. Moreover, we obtain a Bessenrodt-Ono type theorem which gives an inequality of the form $c(a)c(b) > c(a+b)$ for certain families of sequences $c(n)$.

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Kathrin Bringmann, Johann Franke, Bernhard Heim. 2024-01-11. Asymptotics of commuting $\ell$-tuples in symmetric groups and log-concavity. https://arxiv.org/abs/2401.05874

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