Enumerating coprime permutations

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.

11 pages; simplified proof with improved quantitative aspects