On the Number of Permutations Admitting an m-th Root

Abstract

Let $m$ be a positive integer, and $p_n(m)$ the proportion of permutations of the symmetric group $S_n$ that admit an $m$-th root. Calculating the exponential generating function of these permutations, we show the following asymptotic formula $$p_n(m)\, \sim \, {{\pi _m}\over {n^{1-\varphi (m)/m}}},\;\; n\to \infty ,$$ where $\varphi$ is the Euler function and $\pi _m$ an explicit constant.