On the average number of cycles in conjugacy class products

TL;DR

This paper investigates the average number of cycles in the product of two fixed point free permutations, establishing bounds close to the harmonic number, revealing a consistent pattern across conjugacy class products.

Contribution

It provides new bounds on the average number of cycles in products of fixed point free permutations, linking it to harmonic numbers.

Findings

Average number of cycles is between H_n - 3 and H_n + 1.

The bounds are tight and hold for all fixed point free conjugacy classes.

The result applies to random pairs of permutations of given cycle types.

Abstract

We show that for the product of two fixed point free conjugacy classes, the average number of cycles is always very similar. Specifically, our main result is that for a randomly chosen pair of fixed point free permutations of cycle types $\alpha$ and $\beta$, the average number of cycles in their product is between $H_n-3$ and $H_n+1$, where $H_n$ is the harmonic number.