Geodesic
On the number of permutations of special form
Sbornik. Mathematics, Tome 28 (1976) no. 3, pp. 421-429 Cet article a éte moissonné depuis la source Math-Net.Ru

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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 $k<1$ 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. 
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M. P. Mineev; A. I. Pavlov. On the number of permutations of special form. Sbornik. Mathematics, Tome 28 (1976) no. 3, pp. 421-429. http://geodesic.mathdoc.fr/item/SM_1976_28_3_a9/

[1] I. M. Vinogradov, Osnovy teorii chisel, ONTI, Moskva–Leningrad, 1936 | MR

[2] J. Blum, “Enumeration of the squere permutations in $S_n$”, J. Combinatorial theory, (A), 17 (1974), 156–161 | DOI | MR | Zbl

[3] D. Riordan, Vvedenie v kombinatornyi analiz, IL, Moskva, 1963

[4] A. O. Gelfond, Ischislenie konechnykh raznostei, Gostekhizdat, Moskva–Leningrad, 1952 | MR