Permutations and the divisor graph of [1, n]

Abstract

Let $S_{\mathrm{div}}\left(n\right)$ denote the set of permutations π of n such that for each $1⩽j⩽n$ either $j\mid \pi \left(j\right)$ or $\pi \left(j\right)\mid j$. These permutations can also be viewed as vertex-disjoint directed cycle covers of the divisor graph $D_{[1,n]}$ on vertices $v_1,\text{…},v_n$ 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 $c_d={\lim }_{n\to \infty }{(\#S_{\mathrm{div}}\left(n\right))}^{1/n}$ exists and that . We also obtain similar results for the set $S_{\mathrm{lcm}}\left(n\right)$ of permutations where $lcm(j,\pi (j\left)\right)⩽n$ 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.