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

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