Permutations with orders coprime to a given integer

TL;DR

This paper refines the asymptotic understanding of the proportion of permutations with orders coprime to a given integer, providing bounds that depend on Euler's totient function and a constant.

Contribution

It establishes explicit bounds for the proportion of permutations with orders coprime to m, improving upon previous asymptotic results by Pouyanne.

Findings

Bounds depend on Euler's totient function and a constant C(m)

Proportion asymptotically behaves like n^{(φ(m)/m)-1}

Provides explicit inequalities for all n ≥ m

Abstract

Let $m$ be a positive integer and let $\rho(m,n)$ be the proportion of permutations of the symmetric group ${\rm Sym}(n)$ whose order is coprime to $m$. In 2002, Pouyanne proved that $\rho(n,m)n^{1-\frac{\phi(m)}{m}}\sim \kappa_m$ where $\kappa_m$ is a complicated (unbounded) function of $m$. We show that there exists a positive constant $C(m)$ such that, for all $n \geqslant m$, \[C(m) \left(\frac{n}{m}\right)^{\frac{\phi(m)}{m}-1} \leqslant \rho(n,m) \leqslant \left(\frac{n}{m}\right)^{\frac{\phi(m)}{m}-1}\] where $\phi$ is Euler's totient function.