Extremes of generalized inversions on permutation groups

Abstract

Generalized inversions $X_{\mathrm{inv}}^{(d)}$ and generalized descents $X_{\mathrm{des}}^{(d)}$ are an interesting combinatorial extension of the common inversion and descent statistics. By means of the root poset, they can be defined on all classical Weyl groups. In this paper, we investigate the bivariate normality of $(X_{\mathrm{inv}}^{(d)}, X_{\mathrm{des}}^{(d)})^\top$ as well as the extreme value behavior of $X_{\mathrm{inv}}^{(d_1)}$, $X_{\mathrm{des}}^{(d_2)}$ and $(X_{\mathrm{inv}}^{(d_1)}, X_{\mathrm{des}}^{(d_2)})^\top$. We show that bivariate normality holds in the regimes of $d_1 = o(n^{1/3})$ and $d_1 = \omega(n^{1/2})$. For these situations, we also discuss the number of samples $k_n$ for which the Gumbel max-attraction applies to a triangular array based on $X_{\mathrm{inv}}^{(d_1)}$, $X_{\mathrm{des}}^{(d_2)}$ or $(X_{\mathrm{inv}}^{(d_1)}, X_{\mathrm{des}}^{(d_2)})^\top$.