Enumerating coprime permutations

TL;DR

This paper proves a conjecture about the asymptotic count of coprime permutations, revealing a precise exponential growth rate involving an infinite product over primes.

Contribution

It establishes the exact asymptotic number of coprime permutations, confirming a recent conjecture and introducing novel entropy and number-theoretic techniques.

Findings

Number of coprime permutations is asymptotically $n! imes c^n$.

The constant $c$ is an infinite product over primes.

Techniques include entropy maximization and number-theoretic bounds.

Abstract

Define a permutation $\sigma$ to be coprime if $\gcd(m,\sigma(m)) = 1$ for $m\in[n]$. In this note, proving a recent conjecture of Pomerance, we prove that the number of coprime permutations on $[n]$ is $n!\cdot (c+o(1))^n$ where \[c = \prod_{p\text{ prime }}\frac{(p-1)^{2(1-1/p)}}{p\cdot (p-2)^{(1-2/p)}}.\] The techniques involve entropy maximization for the upper bound, and a mixture of number-theoretic bounds, permanent estimates, and the absorbing method for the lower bound.