The R\'enyi entropy of the order of a random permutation

TL;DR

This paper analyzes the Rényi entropy of the order of a random permutation, providing asymptotic results and characterizations for various entropy orders, revealing connections to arithmetic properties of n.

Contribution

It offers the first comprehensive asymptotic analysis of Rényi entropy for permutation orders across all q, including optimal bounds and characterizations for key cases.

Findings

Probability that a random permutation has a specific order is asymptotic to 1/n.

Maximum probability of a permutation's order is asymptotic to 1/n, with characterizations of maximizers.

Probability that two permutations share the same order ranges between c/n^2 and log^*n/n^2.

Abstract

We study the distribution of the order of a random permutation of $[n]$ through the lens of R\'enyi entropy. In particular, we obtain an asymptotic for the R\'enyi $q$-entropy of the order in the full range $1 \leq q \leq \infty$. For $q > 1$, our results are quantitatively optimal and reveal a tight connection between the asymptotic behaviour of the R\'enyi $q$-entropy and arithmetic properties of $n$. Of particular interest are the cases $q = \infty$ and $q = 2$, which correspond to the maximum probability of achieving a particular order and the probability that two independent random permutations have equal orders, respectively. In the former case, we show that the probability in question is asymptotic to $1/n$ and additionally characterise the maximiser for sufficiently large $n$. In the latter case, we determine a minimal and maximal order for the probability as a function of $n$, of respective forms $c/n^2$ and $\log^*n/n^2$. Our results provide an essentially complete answer to a set of questions raised by Acan, Burnette, Eberhard, Schmutz and Thomas, some of which go back to work of Erd\H{o}s and Tur\'an from the 1960s.