  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.
