Permutations With Equal Orders

TL;DR

This paper investigates the probabilities related to two random permutations of [n], showing that the probability they share the same order diminishes as n grows, and that the probability they are conjugate can be significantly larger.

Contribution

The paper proves asymptotic formulas for the probability of permutations sharing the same order and explores the relationship between this probability and conjugacy class probability.

Findings

P(n) = n^{-2+o(1)} as n grows

The ratio P(n)/K(n) can become arbitrarily large

Answers a question posed by Thibault Godin

Abstract

Let $P(n)$ be the probability that two independent, uniformly random permutations of $[n]$ have the same order, and let $K(n)$ be the probability that they are in the same conjugacy class. Answering a question of Thibault Godin, we prove that $ P(n)=n^{-2+o(1)}$ and that $\lim\sup \frac{ P(n) }{ K(n) }=\infty.$