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.
Categories: math.NT math.CO