Fixed points, descents, and inversions in parabolic double cosets of the symmetric group

TL;DR

This paper studies permutation statistics within specific symmetric group subsets, demonstrating asymptotic Poisson and normal distributions for fixed points, descents, and inversions, with explicit convergence rates and concentration results.

Contribution

It introduces new probabilistic limit theorems for permutation statistics in parabolic double cosets, using Stein's method with size-bias coupling and dependency graphs.

Findings

Fixed points follow an asymptotic Poisson distribution.

Descents and inversions are asymptotically normal.

Convergence rates and concentration inequalities are established.

Abstract

We consider statistics on permutations chosen uniformly at random from fixed parabolic double cosets of the symmetric group. We show that the distribution of fixed points is asymptotically Poisson and establish central limit theorems for the distribution of descents and inversions. Our proofs use Stein's method with size-bias coupling and dependency graphs, which also gives convergence rates for our distributional approximations. As applications of our size-bias coupling and dependency graph constructions, we obtain concentration of measure results on the number of fixed points, descents, and inversions.