Mixed r-Stirling numbers of the second kind

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 \( k \geq 1 \) and \( m_i \)’s are positive integers greater than 1.