Permutations with arithmetic constraints

Abstract

Let $S_{\mathrm{lcm}}(n)$ denote the set of permutations $\pi$ of $[n]=\{1,2,\dots ,n\}$ such that $\mathrm{lcm}[j,\pi (j)]\le n$ for each $j\in [n]$. Further, let $S_{\mathrm{div}}(n)$ denote the number of permutations $\pi$ of $[n]$ such that $j\mid \pi (j)$ or $\pi (j)\mid j$ for each $j\in [n]$. Clearly $S_{\mathrm{div}}(n)\subset S_{\mathrm{lcm}}(n)$. We get upper and lower bounds for the counts of these sets, showing they grow geometrically. We also prove a conjecture from a recent paper on the number of "anti-coprime" permutations of $[n]$, meaning that each $\gcd (j,\pi (j))>1$ except when $j=1$.