Fixed points, descents, and inversions in parabolic double cosets of the symmetric group

Abstract

We consider statistics on permutations chosen uniformly at random from fixed parabolic double cosets of the symmetric group. We show that the distribution of fixed points is asymptotically Poisson and establish central limit theorems for the distribution of descents and inversions. Our proofs use Stein's method with size-bias coupling and dependency graphs, which also gives convergence rates for our distributional approximations. As applications of our size-bias coupling and dependency graph constructions, we obtain concentration of measure results on the number of fixed points, descents, and inversions.

Double coset Markov chains

“…For a detailed survey, see [24] or [14]. Double cosets can have very different sizes and [24], [48] develop a probabilistic and enumerative theory. For present applications, an explicit description of the double cosets is needed.…”