Permutations and the divisor graph of $[1,n]$

Let $S_{\mathrm{d}\mathrm{i}\mathrm{v}}(n)$ denote the set of permutations $\pi$ of $n$ such that for each $1\le j\le n$ 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 $v_1,\dots ,v_n$ 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 $c_d={\lim }_{n\to \mathrm{\infty }}{(\mathrm{\#}{S}_{\mathrm{d}\mathrm{i}\mathrm{v}}(n))}^{1\mathrm{/}n}$ exists and that $2.069<c_d<2.694$. We also obtain similar results for the set $S_{\mathrm{l}\mathrm{c}\mathrm{m}}(n)$ of permutations where $\mathrm{l}\mathrm{c}\mathrm{m}(j,\pi (j))\le n$ 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.