Explicit estimates for Comtet numbers of the first kind

Abstract

We give explicit upper and lower bounds for a large subset of Comtet numbers sα(n,m) of the first kind, including the r-Stirling numbers of the first kind, among others. In many occasions, such estimates are asymptotically sharp. The form of the bounds varies according to m lying in the central or non-central regions of {1,…,n}. Depending on this fact, we use different probabilistic representations of sα(n,m) in terms of well known random variables.