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

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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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     author = {A. I. Pavlov},
     title = {On the number of solutions of the equation $x^k=a$ in the symmetric group~$S_n$},
     journal = {Sbornik. Mathematics},
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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/