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 
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/
@article{SM_1981_40_3_a3,
     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},
     pages = {349--362},
     year = {1981},
     volume = {40},
     number = {3},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1981_40_3_a3/}
}
TY  - JOUR
AU  - A. I. Pavlov
TI  - On the number of solutions of the equation $x^k=a$ in the symmetric group $S_n$
JO  - Sbornik. Mathematics
PY  - 1981
SP  - 349
EP  - 362
VL  - 40
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/SM_1981_40_3_a3/
LA  - en
ID  - SM_1981_40_3_a3
ER  - 
%0 Journal Article
%A A. I. Pavlov
%T On the number of solutions of the equation $x^k=a$ in the symmetric group $S_n$
%J Sbornik. Mathematics
%D 1981
%P 349-362
%V 40
%N 3
%U http://geodesic.mathdoc.fr/item/SM_1981_40_3_a3/
%G en
%F SM_1981_40_3_a3

Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[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