Separation probabilities for products of permutations

TL;DR

This paper investigates the probability that specific elements are in distinct cycles in permutations formed by multiplying two random permutations of fixed cycle types, providing exact formulas for these probabilities.

Contribution

It introduces exact formulas for separation probabilities in products of permutations, advancing understanding of their cycle structure and mixing properties.

Findings

Derived an exact formula for the probability that elements are in separate cycles.

Analyzed the cycle structure of permutations formed by product of two fixed cycle type permutations.

Enhanced understanding of mixing properties in permutation groups.

Abstract

We study the mixing properties of permutations obtained as a product of two uniformly random permutations of fixed cycle types. For instance, we give an exact formula for the probability that elements $1,2,...,k$ are in distinct cycles of the random permutation of $\{1,2,...,n\}$ obtained as product of two uniformly random $n$-cycles.