Geodesic
On the number of solutions of the equation $x^k=a$ in the symmetric group $S_n$
Sbornik. Mathematics, Tome 40 (1981) no. 3, pp. 349-362 Cet article a éte moissonné depuis la source Math-Net.Ru

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This paper consists of three sections. In the first a formula is given for the number $N^{(k)}_n(a)$ of solutions of the equation $x^k=a$ in $S_n$ depending on the cyclic structure of the permutation $a$. In the second an asymptotic formula is given for the quantity $M^{(k)}_n=\max_{a\in S_n}N^{(k)}_n(a)$ for a fixed $k\geqslant2$ as $n\to\infty$. In the third an asymptotic formula is found for the cardinality of the set of permutations $a$ such that the equation $x^k=a$ has a unique solution. Bibliography: 5 titles. 
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     title = {On the number of solutions of the equation $x^k=a$ in the symmetric group~$S_n$},
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     pages = {349--362},
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     url = {http://geodesic.mathdoc.fr/item/SM_1981_40_3_a3/}
}
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A. I. Pavlov. On the number of solutions of the equation $x^k=a$ in the symmetric group $S_n$. Sbornik. Mathematics, Tome 40 (1981) no. 3, pp. 349-362. http://geodesic.mathdoc.fr/item/SM_1981_40_3_a3/

[1] M. P. Mineev, A. I. Pavlov, “O chisle podstanovok spetsialnogo vida”, Matem. sb., 99(141) (1976), 468–476 | MR | Zbl

[2] M. A. Evgrafov, Asimptoticheskie otsenki i tselye funktsii, Fizmatgiz, Moskva, 1962 | MR

[3] V. N. Sachkov, Veroyatnostnye metody v kombinatornom analize, izd-vo “Nauka”, Moskva, 1978 | MR

[4] E. Titchmarsh, Teoriya funktsii, Gostekhizdat, Moskva–Leningrad, 1951

[5] E. A. Bender, “Asymptotic methods in enumeration”, SIAM Review, 16:4 (1974), 485–515 | DOI | MR | Zbl