Counting inversions and descents of random elements in finite Coxeter groups

TL;DR

This paper studies the distributions of inversions and descents in finite Coxeter groups, providing formulas for their means, variances, and limit theorems, thus advancing the understanding of permutation statistics in algebraic structures.

Contribution

It offers uniform formulas for means, variances, and limit theorems for Mahonian and Eulerian distributions in finite Coxeter groups, extending classical permutation statistics.

Findings

Derived formulas for means and variances in Coxeter groups.

Established conditions for central and local limit theorems.

Analyzed the distribution of descents and inversions in various Coxeter groups.

Abstract

We investigate Mahonian and Eulerian probability distributions given by inversions and descents in general finite Coxeter groups. We provide uniform formulas for the means and variances in terms of Coxeter group data in both cases. We also provide uniform formulas for the double-Eulerian probability distribution of the sum of descents and inverse descents. We finally establish necessary and sufficient conditions for general sequences of Coxeter groups of increasing rank under which Mahonian and Eulerian probability distributions satisfy central and local limit theorems.