Mixed r-stirling numbers of the second kind

Abstract

The Stirling number of the second kind $\left\{\genfrac{}{}{0ex}{}{n}{k}\right\}$ counts the number of ways to partition a set of $n$ labeled balls into $k$ non-empty unlabeled cells. As an extension of this, we consider $b_1+b_2+\dots +b_n$ balls with $b_1$ balls labeled $1$, $b_2$ balls labeled $2$, $\dots$, $b_n$ balls labeled $n$ and $c_1+c_2+\dots +c_k$ cells with $c_1$ cells labeled $1$, $c_2$ cells labeled $2$, $\dots$, $c_k$ cells labeled $k$ and then we called the number of ways to partition the set of these balls into non-empty cells of these types as the \textit{mixed partition numbers}. As an application, we give a new statement of the $r$-Stirling numbers of second kind and $r$-Bell numbers. We also introduce the \textit{$r$-mixed Stirling number of second kind and $r$-mixed Bell numbers}. Finally, for a positive integer $m$ we evaluate the number of ways to write $m$ as the form $m_1\cdot m_2\cdot \dots \cdot m_k$, where $k⩾1$ and $m_i$'s are positive integers greater than $1$.