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

TL;DR

This paper studies permutations related to divisor and least common multiple conditions, establishing growth rates and limits for their counts, and introduces a graph-theoretic bound on cycle covers that could be useful beyond this context.

Contribution

It proves the existence of the growth rate limit for divisor-permutations and bounds it, also extending results to lcm-permutations, with a new graph-theoretic bound of independent interest.

Findings

Limit of the number of divisor-permutations exists and is between 2.069 and 2.694.

Similar bounds are obtained for lcm-permutations.

A new graph-theoretic bound on directed cycle covers is established.

Abstract

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.