Asymptotics of commuting ℓ -tuples in symmetric groups and log-concavity

Abstract

Denote by $N_{\ell }\left(n\right)$ 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 $N_{\ell }\left(n\right)$ . In addition, general criteria for log-concavity are shown, which can be applied to $N_{\ell }\left(n\right)$ among other examples. Moreover, we obtain a Bessenrodt-Ono type theorem which gives an inequality of the form $c\left(a\right)c\left(b\right)>c(a+b)$ for certain families of sequences c(n).

Keywords: Generating functions; Log-concavity; Partition numbers; Symmetric group.