Permutations and the divisor graph of [1, <i>n</i> ]

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