The most probable order of a random permutation

TL;DR

This work investigates the local behavior of the order $\mathrm{ord}(\pi_n)$ of a uniformly random permutation on $[n]$ and identifies the most probable order. It develops an anticoncentration analysis by partitioning on the number of cycles and using a recursion for $p_n(m)$ along with sharp cycle-count bounds, showing $M(n) = \max_m p_n(m) \sim 1/n$ and that the maximum is attained at $m=n-\max K_n$, where $K_n=\{k: \operatorname{lcm}(1,\dots,k) \mid n-k\}$. A precise local expansion is obtained for $\mathbb{P}(\mathrm{ord}(\pi_n)=n-k)$ with $k\in K_n$, namely $\mathbb{P}(\mathrm{ord}(\pi_n)=n-k)=\frac{1}{n-k}+\eta(n,k)+O(n^{-3+o(1)})$, leading to $M(n)=\frac{1}{n}+O\left(\frac{\log n}{n^2}\right)$. The results resolve questions attributed to Erdős-Turán and Acan et al. about the mode and concentration of $\mathrm{ord}(\pi_n)$ and provide a detailed structural description of near-maximum orders.

Abstract

Given positive integers $n$ and $m$, let $p_n(m)$ be the probability that a uniform random permutation of $[n]$ has order exactly $m$. We show that, as $n \to \infty$, the maximum of $p_n(m)$ over all $m$ is asymptotic to $1/n$, the probability of an $n$-cycle. Furthermore, for sufficiently large $n$, we show that the maximum is attained precisely if $m$ is the least positive integer divisible by all positive integers less than or equal to $n-m$. This answers a question of Acan, Burnette, Eberhard, Schmutz and Thomas, originally attributed to work of Erdős and Turán from 1968.

Theorems & Definitions (14)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Corollary 2.2
• Lemma 2.3
• Corollary 2.4
• Lemma 2.5
• proof : Proof sketch.
• Proposition 3.1
• proof