How to write a permutation as a product of involutions (and why you might care)

TL;DR

This paper provides an explicit formula for counting the ways to express any permutation as a product of two involutions, depending on its cycle structure, revealing connections to symmetric group characters.

Contribution

It introduces a formula for the number of involution factorizations of permutations based on cycle type, linking combinatorics and representation theory.

Findings

Number of involution factorizations depends only on cycle type

In many cases, these counts relate to irreducible characters of the symmetric group

Provides explicit formulas and explores their algebraic significance

Abstract

It is well-known that any permutation can be written as a product of two involutions. We provide an explicit formula for the number of ways to do so, depending only on the cycle type of the permutation. In many cases, these numbers are sums of absolute values of irreducible characters of the symmetric group evaluated at the same permutation, although apart from the case where all cycles are the same size, we have no good explanation for why this should be so.