  Let $S_{\rm div}(n)$ denote the set of permutations $\pi$ of $n$ such that
for each $1\leq j \leq n$ either $j \mid \pi(j)$ or $\pi(j) \mid j$. These
permutations can also be viewed as vertex-disjoint directed cycle covers of the
divisor graph $\mathcal{D}_{[1,n]}$ on vertices $v_1, \ldots, v_n$ with an edge
between $v_i$ and $v_j$ if $i\mid j$ or $j \mid i$. We improve on recent
results of Pomerance by showing $c_d = \lim_{n \to \infty }\left(\# S_{\rm
div}(n)\right)^{1/n}$ exists and that $2.069<c_d<2.694$. We also obtain similar
results for the set $S_{\rm lcm}(n)$ of permutations where ${\rm
lcm}(j,\pi(j))\leq n$ for all $j$. The results rely on a graph theoretic result
bounding the number of vertex-disjoint directed cycle covers, which may be of
independent interest.
