Extreme Values of Permutation Statistics

TL;DR

This paper studies the extreme values of Mahonian and Eulerian permutation statistics from Coxeter groups, using large deviations theory to establish Gumbel distribution convergence under specific bounds.

Contribution

It introduces a framework for analyzing extreme permutation statistics in Coxeter groups and proves Gumbel attraction results with bounds on sample sizes.

Findings

Gumbel distribution attracts the extreme values of Mahonian and Eulerian distributions.

Different bounds on sample size $k_n$ are necessary for each distribution class.

The approach applies large deviations theory to permutation statistics in Coxeter groups.

Abstract

We investigate extreme values of Mahonian and Eulerian distributions arising from counting inversions and descents of random elements of finite Coxeter groups. To this end, we construct a triangular array of either distribution from a sequence of Coxeter groups with increasing ranks. To avoid degeneracy of extreme values, the number of i.i.d. samples $k_n$ in each row must be asymptotically bounded. We employ large deviations theory to prove the Gumbel attraction of Mahonian and Eulerian distributions. It is shown that for the two classes, different bounds on $k_n$ ensure this.