On the number of permutations of special form

M. P. Mineev ; A. I. Pavlov

Sbornik. Mathematics, Tome 28 (1976) no. 3, pp. 421-429

Résumé

It is shown that the number of permutations $a$ for which the equation $x^k=a$, where $a\in S_n$ ($S_n$ is the symmetric group of degree $n$) and $k1$ is a fixed natural number, has at least one solution $x\in S_n$ is asymptotically equal to $$ C(k)n^{\varphi(k)/k-1/2}\biggl(\frac ne\biggr)^n\quad\text{as}\quad n\to\infty, $$ where $C(k)$ is a constant depending only on $k$, and $\varphi(k)$ is the Euler function. Bibliography: 4 titles.