A Central Limit Theorem on Two-Sided Descents of Mallows Distributed Elements of Finite Coxeter Groups

The Mallows distribution is a non-uniform distribution, first introduced over permutations to study non-ranked data, in which permutations are weighted according to their length. It can be generalized to any Coxeter group, and we study the distribution of des(w) + des(w⁻¹) where w is a Mallows distributed element of a finite irreducible Coxeter group. We show that the asymptotic behavior of this statistic is Guassian. The proof uses a size-bias coupling with Stein's method.