Geodesic
Random permutations with cycle lengths in a~given finite set
Diskretnaya Matematika, Tome 20 (2008) no. 1, pp. 25-37.

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We consider the class of all permutations of degree $n$ whose cycle lengths are elements of a fixed finite set $A\subset\mathbf N$ such that $\operatorname{card}A\ge2$ and $\operatorname{gcd}\{k\mid k\in A\}=1$. Under the assumption that the permutation $X$ is equiprobably chosen from this class, we obtain a multidimensional local normal theorem for the joint distribution of the numbers of cycles of given sizes in this permutation. The obtained results are utilised and sharpened in the case where $X$ is an equiprobably chosen solution of the equation $X^r=e$, where $e$ is an identity permutation of degree $n$, $r\ge2$ is a fixed positive integer. 
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A. N. Timashev. Random permutations with cycle lengths in a~given finite set. Diskretnaya Matematika, Tome 20 (2008) no. 1, pp. 25-37. http://geodesic.mathdoc.fr/item/DM_2008_20_1_a1/

[1] Chowla S., Herstein I. N., Scott W. R., “The solution of $x^d=1$ in symmetric groups”, Norske Vid. Selsk. Forhdl., 25 (1952), 29–31 | MR | Zbl

[2] Kolchin V. F., Sluchainye grafy, Fizmatlit, Moskva, 2000 | MR | Zbl

[3] Bender E. A., “Asimptoticheskie metody v teorii perechislenii”, Perechislitelnye zadachi kombinatornogo analiza, Mir, Moskva, 1979, 266–310

[4] Volynets L. M., “Chislo reshenii uravneniya $x^s=e$ v simmetricheskoi gruppe”, Matem. zametki, 40:2 (1986), 586–589 | MR | Zbl

[5] Pavlov A. I., “O podstanovkakh s dlinami tsiklov iz zadannogo mnozhestva”, Teoriya veroyatnostei i ee primeneniya, 31:3 (1986), 618–619

[6] Wilf H., “The asymptotics of $e^{P(x)}$ and the number of elements of each order in $S_n$”, Bull. Amer. Math. Soc., 15 (1986), 228–232 | DOI | MR | Zbl

[7] Moser L., Wyman M., “On the solution of $x^d=1$ in symmetric groups”, Canad. J. Math., 7 (1955), 159–168 | MR | Zbl

[8] Volynets L. M., “Chislo reshenii odnogo uravneniya v simmetricheskoi gruppe”, Veroyatnostnye protsessy i ikh prilozheniya, MIEM, Moskva, 1985, 104–109

[9] Timashev A. N., “Predelnye teoremy v skhemakh razmescheniya chastits po razlichnym yacheikam s ogranicheniyami na zapolneniya yacheek”, Teoriya veroyatnostei i ee primeneniya, 49:4 (2004), 712–725 | MR

[10] Sachkov V. N., “Asimptoticheskie formuly i predelnye raspredeleniya dlya kombinatornykh konfiguratsii, porozhdaemykh mnogochlenami”, Diskretnaya matematika, 19:3 (2007), 3–14 | MR | Zbl