A Central Limit Theorem on Two-Sided Descents of Mallows Distributed Elements of Finite Coxeter Groups

TL;DR

This paper establishes a Gaussian limit law for the sum of descents of an element and its inverse in Mallows distributed Coxeter group elements, using Stein's method and size-bias coupling.

Contribution

It extends the central limit theorem to two-sided descent statistics in Mallows distributions over finite Coxeter groups, a novel generalization.

Findings

Asymptotic Gaussian behavior of the descent sum

Application of Stein's method with size-bias coupling

Generalization to finite Coxeter groups

Abstract

The Mallows distribution is a non-uniform distribution, first introduced over permutations to study non-ranked data, in which permutations are weighted according to their length. It can be generalized to any Coxeter group, and we study the distribution of $\text{des}(w) + \text{des}(w^{-1})$ where $w$ is a Mallows distributed element of a finite irreducible Coxeter group. We show that the asymptotic behavior of this statistic is Guassian. The proof uses a size-bias coupling with Stein's method.