Permutations With Equal Orders
Huseyin Acan, Charles Burnette, Sean Eberhard, Eric Schmutz, James, Thomas

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 be the probability that two independent, uniformly random permutations of have the same order, and let be the probability that they are in the same conjugacy class. Answering a question of Thibault Godin, we prove that and that
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