Poisson approximation for large permutation groups

TL;DR

This paper studies the probabilistic behavior of large permutation groups formed by block permutations, deriving novel limit theorems showing that various permutation statistics converge to compound Poisson distributions with complex dependence.

Contribution

It introduces new limit theorems for permutation statistics in block permutation groups, revealing their convergence to compound Poisson distributions with dependence structures.

Findings

Limit theorems for fixed points and cycle counts

Convergence to compound Poisson distributions

Dependence structures in permutation statistics

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/or enumerative aspects of random elements of $G_{k,n}$. This includes novel limit theorems for fixed points, cycles of various lengths, number of cycles and inversions. The limits are compound Poisson distributions with interesting dependence structure.