Joint extremes of inversions and descents of random permutations

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.