Poisson approximation for large permutation groups

Abstract

Let $G_{k,n}$ be a group of permutations of kn objects which permutes things independently in disjoint blocks of size k and then permutes the blocks. We investigate the probabilistic and enumerative aspects of random elements of $G_{k,n}$. This includes novel limit theorems for cycles of various lengths, number of cycles, and inversions. The limits include compound Poisson distributions with interesting dependence structure.