Coprime permutations

TL;DR

This paper investigates permutations where each position and its image are coprime, establishing bounds on their count for large n, revealing asymptotic growth rates between n!/3.73^n and n!/2.5^n.

Contribution

The paper provides new bounds on the number of coprime permutations, advancing understanding of their asymptotic behavior for large n.

Findings

C(n) is bounded between n!/3.73^n and n!/2.5^n for large n

The bounds improve understanding of the distribution of coprime permutations

Asymptotic growth rate of coprime permutations is characterized

Abstract

Let $C(n)$ denote the number of permutations $\sigma$ of $[n]=\{1,2,\dots,n\}$ such that $\gcd(j,\sigma(j))=1$ for each $j\in[n]$. We prove that for $n$ sufficiently large, $n!/3.73^n < C(n) < n!/2.5^n$.