Joint extremes of inversions and descents of random permutations

TL;DR

This paper develops asymptotic theory for the joint distribution of inversions and descents in random permutations, showing they converge to a bivariate Gumbel distribution with independent margins, and extends results to Coxeter groups.

Contribution

It introduces a novel Gaussian approximation approach to analyze the dependency between permutation statistics and extends the theory to Coxeter groups.

Findings

Joint distribution converges to a bivariate Gumbel distribution.

Provides the first CLT for permutation statistics in Coxeter groups.

Shows independence of margins in the limiting distribution.

Abstract

We provide asymptotic theory for the joint distribution of $X_{\mathrm{inv}}$ and $X_{\mathrm{des}}$, the numbers of inversions and descents of random permutations. Recently, D\"orr & Kahle (2022) proved that $X_{\mathrm{inv}}$, respectively, $X_{\mathrm{des}}$ is in the maximum domain of attraction of the Gumbel distribution. To tackle the dependency between these two permutation statistics, we use H\'ajek projections and a suitable quantitative Gaussian approximation. We show that $(X_{\mathrm{inv}}, X_{\mathrm{des}})$ is in the maximum domain of attraction of the two-dimensional Gumbel distribution with independent margins. This result can be stated in the broader combinatorial framework of finite Coxeter groups, on which our method also yields the central limit theorem for $(X_{\mathrm{inv}}, X_{\mathrm{des}})$ and various other permutation statistics as a novel contribution. In particular, signed permutation groups with random biased signs and products of classical Weyl groups are investigated.