On the number of permutations admitting an \(m\)th root (Q5954320)

Summary: 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 \[ p_n(m)\sim {{\pi _m}\over {n^{1-\varphi (m)/m}}},\quad n\to \infty , \] where \(\varphi\) is the Euler function and \(\pi _m\) an explicit constant.