Mixed r-Stirling numbers of the second kind

Abstract

The Stirling number of the second kind $S(n,k)$ counts the number of ways to partition a set of $n$ labeled balls into $k$ non-empty unlabeled cells. We extend this problem and give a new statement of the $r$-Stirling numbers of the second kind and $r$-Bell numbers. We also introduce the $r$-mixed Stirling number of the second kind and $r$-mixed Bell numbers. As an application of our results we obtain a formula for the number of ways to write an integer $m>0$ in the form $m_1\cdot m_2\cdot \cdots \cdot m_k$, where $k\ge 1$ and $m_i$’s are positive integers greater than 1.