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

Denote by Nℓ(n) the number of ℓ -tuples of elements in the symmetric group Sn with commuting components, normalized by the order of Sn . In this paper, we prove asymptotic formulas for Nℓ(n) . In addition, general criteria for log-concavity are shown, which can be applied to Nℓ(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).