Mixed r-stirling numbers of the second kind

TL;DR

This paper introduces mixed partition numbers generalizing Stirling numbers of the second kind, explores their properties, and applies them to define new $r$-mixed Stirling and Bell numbers, along with an integer factorization problem.

Contribution

It extends classical Stirling and Bell numbers to mixed labeled partitions and introduces $r$-mixed variants, providing new combinatorial identities and applications.

Findings

Defined mixed partition numbers for labeled balls and cells.

Established new formulas for $r$-mixed Stirling and Bell numbers.

Evaluated the number of factorizations of integers into products greater than one.

Abstract

The Stirling number of the second kind $n\brace k$ 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 $b_1+b_2+\ldots+b_n$ balls with $b_1$ balls labeled $1$, $b_2$ balls labeled $2$, $\ldots$, $b_n$ balls labeled $n$ and $c_1+c_2+\ldots+c_k$ cells with $c_1$ cells labeled $1$, $c_2$ cells labeled $2$, $\ldots$, $c_k$ 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 $m_1\cdot m_2\cdot\ldots\cdot m_k$, where $k\geqslant 1$ and $m_i$'s are positive integers greater than $1$.