Asymptotics of commuting $\ell$-tuples in symmetric groups and log-concavity
Kathrin Bringmann, Johann Franke, Bernhard Heim
TL;DR
This work analyzes the asymptotics of commuting $\ell$-tuples in the symmetric group, encapsulated by $N_{\ell}(n)=|C_{\ell,n}|/|S_n|$, and derives a unified multi-pole saddle-point framework that yields explicit large-$n$ expansions for all $\ell$, with particularly sharp forms for $\ell\in\{2,3,4,5\}$ and a general formula for $\ell\ge6$. It establishes a general log-concavity criterion for sequences with leading exponential growth and demonstrates log-concavity for many natural arithmetic-counting families, including $N_{\ell}(n)$ in broad regimes, via a careful comparison of leading exponential terms. The paper also proves a Bessenrodt–Ono type inequality $c(a)c(b) > c(a+b)$ for large arguments in these families, which implies corresponding supermultiplicativity-type inequalities for $N_{\ell}(n)$. Collectively, these results advance understanding of partition-like counting in permutation groups, connect to orbifold/Euler-characteristic interpretations, and provide a toolkit for establishing log-concavity in complex generating-function settings. The methods leverage the BBBF23 framework, Dirichlet-series pole structures, and Lagrange inversion to extract precise asymptotics and correction terms that govern growth and concavity properties.
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 $c(n)$.