# Some Connections Between Cycles and Permutations that Fix a Set and Touchard Polynomials and Covers of Multisets

## TL;DR

This paper explores the relationship between permutation cycles, Touchard polynomials, and multiset covers, introducing a new permutation statistic and providing a simpler derivation of related generating functions.

## Contribution

It offers a novel proof linking permutation cycles to Touchard polynomials and introduces a new permutation statistic for fixed set counts.

## Key findings

- New proof connecting permutation cycles and Touchard polynomials.
- Introduction of a permutation statistic counting fixed sets.
- Simplified derivation of generating functions for multiset covers.

## Abstract

We present a new proof of a fundamental result concerning cycles of random permutations which gives some intuition for the connection between Touchard polynomials and the Poisson distribution. We also introduce a rather novel permutation statistic and study its distribution. This quantity, indexed by $m$, is the number of sets of size $m$ fixed by the permutation. This leads to a new and simpler derivation of the exponential generating function for the number of covers of certain multisets.