Explicit estimates for the Stirling numbers of the second kind

We give explicit estimates for the Stirling numbers of the second kind $S(n,m)$. With a few exceptions, such estimates are asymptotically sharp. The form of these estimates varies according to m lying in the central or non-central regions of \1,… ,n\. In each case, we use a different probabilistic representation of $S(n,m)$ in terms of well known random variables to show the corresponding results.