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

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.