# Mixed coloured permutations

## TL;DR

This paper introduces mixed coloured permutations with coloured cycles, deriving their enumeration formulas, recursions, and identities, and explores their connections to Stirling numbers and Bell polynomials.

## Contribution

It presents the concept of mixed coloured permutations and provides new enumerative formulas and identities, extending classical Stirling numbers.

## Key findings

- Derived generating functions and closed-form formulas.
- Established recursions and combinatorial identities.
- Connected mixed Stirling numbers to Bell polynomials.

## Abstract

In this paper we introduce mixed coloured permutation, permutations with certain coloured cycles, and study the enumerative properties of these combinatorial objects. We derive the generating function, closed forms, recursions and combinatorial identities for the counting sequence, mixed Stirling numbers of the first kind. In this comprehensive study we consider further the conditions on the length of the cycles, $r$-mixed Stirling numbers and the connection to Bell polynomials.