On the number of mth roots of permutations

TL;DR

This paper derives explicit formulas and generating functions for counting mth roots of permutations, analyzes their probabilities, and proves a uniformity property for these probabilities when m is a prime power.

Contribution

It provides a new explicit expression and generating function for the number of mth roots of permutations, along with probabilistic analysis and a novel uniformity result.

Findings

Explicit formula for the number of mth roots of a permutation.

Generating function for counting mth roots.

Probability distribution of mth roots in random permutations.

Abstract

Let m be a fixed positive integer. It is well-known that a permutation $\sigma$ may have one, many, or no mth roots. In this note we provide an explicit expression and a generating function for the number of mth roots of \sigma. Let p_m(n) be the probability that a random n-permutation has an mth root. We also include a proof that p_m(jq)=p_m(jq+1)=... =p_m(jq+(q-1)) where j=0,1,... and m is a power of prime q.