On the uniformity of the approximation for $r$-associated Stirling numbers of the second Kind

Abstract

The $r$-associated Stirling numbers of the second kind are a natural extension of Stirling numbers of the second kind. A combinatorial interpretation of $r$-associated Stirling numbers of the second kind is the number of ways to partition $n$ elements into $m$ subsets such that each subset contains at least $r$ elements. Calculating the associated Stirling numbers is typically done with a recurrence relation or a generating function that are computationally expensive or alternatively with a closed-form that is practical for only a limited parameter range. In 1994 Hennecart proposed an approximation for the $r$-associated Stirling numbers that is fast to compute, is amenable to analysis over a wide range of parameters, and is conjectured to be asymptotically tight. There are a few other approximations for the associated Stirling numbers, but none of them are as general as Hennecart's. However, until this work, Hennecart's approximation had been utilized without a proper justification due to the absence of a rigorous proof. This work provides a proof of the uniformity of the Hennecart approximation.