  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.
