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

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).