A chord diagram is a set of n chords joining distinct endpoints on a circle. We study the asymptotic distributions of the number of crossings and the number of simple chords in a random chord diagram. Using size-bias coupling and Stein's method, we obtain bounds on the Wasserstein distance between the distribution of the number of crossings and a standard normal. We also give upper and lower bounds of the same order on the total variation distance between the distribution of the number of simple chords and a Poisson random variable.

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.

We consider a long-range growth dynamics on the two-dimensional integer lattice, initialized by a finite set of occupied points. Subsequently, a site x becomes occupied if the pair consisting of the counts of occupied sites along the entire horizontal and vertical lines through x lies outside a fixed Young diagram Z. We study the extremal quantity µ(Z), the maximal finite time at which the lattice is fully occupied. We give an upper bound on µ(Z) that is linear in the area of the bounding rectangle of Z, and a lower bound √ s − 1, where s is the side length of the largest square contained in Z. We give more precise results for a restricted family of initial sets, and for a simplified version of the dynamics.

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