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

Denote by $N_{\mathrm{\ell }}(n)$ the number of $\mathrm{\ell }$-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 $N_{\mathrm{\ell }}(n)$. In addition, general criteria for log-concavity are shown, which can be applied to $N_{\mathrm{\ell }}(n)$ 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 $c(n)$.