The generating functions of stirling numbers of the second kind derived probabilistically

Stirling numbers of the second kind, S(n, r), denote the number of partitions of a finite set of size n into r disjoint nonempty subsets. The aim of this short article is to shed some light on the generating functions of these numbers by deriving them probabilistically. We do this by linking them to Markov chains related to the classical coupon collector problem; coupons are collected in discrete time (ordinary generating function) or in continuous time (exponential generating function). We also review the shortest possible combinatorial derivations of these generating functions.