Joint Probabilities within Random Permutations

TL;DR

This paper explores the analogy between prime factorizations and permutation cycle decompositions, providing asymptotic formulas and methods for computing related probabilities in the context of random permutations.

Contribution

It extends asymptotic formulas from number theory to permutation cycle structures and surveys practical probability computation methods.

Findings

Asymptotic formulas for semismooth numbers apply to permutations.

Practical methods for computing multivariate probabilities are presented.

The prime permutation analogy offers new insights into cycle structure analysis.

Abstract

A celebrated analogy between prime factorizations of integers and cycle decompositions of permutations is explored here. Asymptotic formulas characterizing semismooth numbers (possessing at most several large factors) carry over to random permutations. We offer a survey of practical methods for computing relevant probabilities of a bivariate or trivariate flavor.