arXiv 2207.096522022-07math.NTmath.CO

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

Nathan McNew

英文摘要：

Let $S_{\rm div}(n)$ denote the set of permutations $π$ of $n$ such that for each $1\leq j \leq n$ either $j \mid π(j)$ or $π(j) \mid j$. 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 ${\rm lcm}(j,π(j))\leq n$ 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.

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