A counting version of Petersen's $2$-factor theorem

A classical result of Petersen states that every regular graph of even degree has a $2$-factor. We prove that every $n$-vertex $2r$-regular simple graph contains at least ${((1+o_r(1))\frac{2r}{e})}^n$ distinct $2$-factors. This improves the previously known lower bound ${((1+o_r(1))\frac{r}{e})}^n$ by a factor of $2^{(1+o(1))n}$ and is asymptotically tight for large $r$. As a direct consequence, we determine asymptotically tight bounds on the number of $2$-factorizations of a given $2r$-regular simple graph for every sufficiently large $r$.