Permutation statistics of products of random permutations

TL;DR

This paper develops a method to compute the expected values of various permutation statistics on products of random permutations, using character theory and class functions, providing explicit formulas for common statistics.

Contribution

It introduces a framework to calculate expected permutation statistics on products of random permutations using character decompositions, extending prior methods.

Findings

Derived formulas for expected excedance, inversion, descent, major index, and k-cycle statistics.

Provided explicit character decompositions for these statistics.

Enabled calculation of expectations for products of permutations from conjugacy classes.

Abstract

Given a permutation statistic $s : S_n \to \mathbb{R}$, define the mean statistic $\bar{s}$ as the statistic which computes the mean of $s$ over conjugacy classes. We describe a way to calculate the expected value of $s$ on a product of $t$ independently chosen elements from the uniform distribution on a union of conjugacy classes $\Gamma \subseteq S_n$. In order to apply the formula, one needs to express the class function $\bar{s}$ as a linear combination of irreducible $S_n$-characters. We provide such expressions for several commonly studied permutation statistics, including the excedance number, inversion number, descent number, major index and $k$-cycle number. In particular, this leads to formulae for the expected values of said statistics.