# Mixed restricted Stirling numbers

## TL;DR

This paper explores mixed restricted Stirling numbers, providing formulas, generating functions, and recurrences, with a focus on a special case linked to r-Stirling numbers, using combinatorial and symbolic methods.

## Contribution

It introduces new formulas and generating functions for mixed restricted Stirling numbers, including a special case connected to r-Stirling numbers, advancing combinatorial understanding.

## Key findings

- Derived a general formula for mixed restricted Stirling numbers
- Established generating functions and recurrences for a special case
- Connected the special case to r-Stirling numbers

## Abstract

In this note we investigate mixed partitions with extra condition on the sizes of the blocks. We give a general formula and the generating function. We consider in more details a special case, determining the generating functions, some recurrences and a connection to r-Stirling numbers. To obtain our results, we use pure combinatorial arguments, classical manipulations of generating functions and to derive the generating functions we apply the symbolic method.