# On the number of even roots of permutations

## TL;DR

This paper derives generating functions for counting even and odd k-th roots of permutations, connecting to known results and OEIS sequences, thus advancing understanding of permutation roots and their parity properties.

## Contribution

It provides new generating functions for counting even and odd k-th roots of permutations, extending previous results and linking to OEIS sequences.

## Key findings

- Derived explicit generating functions for even and odd k-th roots
- Connected new results to known OEIS sequences
- Enhanced understanding of permutation root structures

## Abstract

Let $\sigma$ be a permutation on $n$ letters. We say that a permutation $\tau$ is an even (resp. odd) $k$th root of $\sigma$ if $\tau^k=\sigma$ and $\tau$ is an even (resp. odd) permutation. In this article, we obtain generating functions for the number of even and odd $k$th roots of permutations. Our result implies know generating functions of Moser and Wyman and also some generating functions for sequences in The On-line Encyclopedia of Integer Sequences (OEIS).