Geodesic
Asymptotics with remainder term for moments of the total cycle number of random $A$-permutation
Diskretnaya Matematika, Tome 31 (2019) no. 3, pp. 114-127.

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We consider the set of $n$-permutations with cycle lengths belonging to some fixed set $A$ of natural numbers (so-called $A$-permutations). Let random permutation $\tau_n$ be uniformly distributed on this set. For some class of sets $A$ we find the asymptotics with remainder term for moments of total cycle number of $\tau_n$.
Keywords: random $A$-permutations, the total number of cycles, the number of cycles of fixed length. 
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A. L. Yakymiv. Asymptotics with remainder term for moments of the total cycle number of random $A$-permutation. Diskretnaya Matematika, Tome 31 (2019) no. 3, pp. 114-127. http://geodesic.mathdoc.fr/item/DM_2019_31_3_a7/

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

[2] Goncharov V.L., “Iz oblasti kombinatoriki”, Izv. AN SSSR, Ser. matem., 8 (1944), 3–48 | Zbl

[3] Zacharovas V., “Distribution of the logarithm of the order of a random permutation”, Lith. Math. J., 44:3 (2004), 296–327 | DOI | MR | Zbl

[4] Klimavicius, J., Manstavicius, E., “Turan–Kubilius' inequality on permutations”, Ann. Univ. Sci. Budapest. Sect. Comput., 48 (2018), 45–51 | MR | Zbl

[5] Manstavicius E., Stepas V., “Variance of additive functions defined on random assemblies”, Lith. Math. J., 57:2 (2017), 222–235 | DOI | MR | Zbl

[6] Pavlov A. I., “On the limit distribution of the number of cycles and the logarithm of the order of a class of permutations”, Math. USSR-Sb., 42:4 (1982), 539–567 | DOI | MR | MR | Zbl

[7] A. I. Pavlov, “On some classes of permutations with number-theoretic restrictions on the lengths of cycles”, Math. USSR-Sb., 57:1 (1987), 263–275 | DOI | MR | Zbl | Zbl

[8] Pavlov A. I., “On the number of substitutions with cycle lengths from a given set”, Discrete Math. Appl., 2:4 (1992), 445–459 | DOI | MR | Zbl

[9] Pavlov A. I., “Arithmetic problems of combinatorial analysis”, Math. Notes, 55:2 (1994), 173–177 | DOI | MR | Zbl

[10] Pavlov A. I., “On two classes of permutations with number-theoretic conditions on the lengths of the cycles”, Math. Notes, 62:6 (1997), 739–746 | DOI | DOI | MR | Zbl

[11] Postnikov A. G., Introduction to Analytic Number Theory, Amer. Math. Soc., 1988, 320 pp. | MR | Zbl

[12] Sachkov V. N., “Mappings of a finite set with limitations on contours and height”, Theory Probab. Appl., 17:4 (1973), 640–656 | DOI | MR | Zbl

[13] Sachkov V. N., “Random mappings with bounded height”, Theory Probab. Appl., 18:1 (1973), 120–130 | DOI | MR | Zbl

[14] Sachkov V. N., Combinatorial Methods in Discrete Mathematics, Cambridge University Press, 1996, 317 pp. | MR

[15] Sachkov V. N., Probabilistic Methods in Combinatorial Analysis, Cambridge University Press, 1997, 246 pp. | MR | Zbl

[16] Sachkov V.N., Vvedenie v kombinatornye metody diskretnoi matematiki, MTsNMO, M., 2004

[17] Seneta E., Regularly Varying Functions, Lecture Notes in Mathematics, 508, Springer-Verlag, 1976, 116 pp. | DOI | MR | Zbl

[18] Storm J., Zeindler D., “Total variation distance and the Erdős–Turán law for random permutations with polynomially growing cycle weights”, Ann. Inst. H. Poincaré Probab. Stat., 52:4 (2016), 1614–1640 | DOI | MR | Zbl

[19] Yakymiv A. L., Probabilistic applications of Tauberian theorems, Modern Probability and Statistics, VSP, Leiden, 2005, viii+225 pp. | MR | Zbl

[20] Yakymiv A. L., “On the distribution of the $m$th maximal cycle lengths of random A-permutations”, Discrete Math. Appl., 15:5 (2005), 527–546 | DOI | DOI | MR | Zbl

[21] Yakymiv A. L., “A limit theorem for the logarithm of the order of a random $A$-permutation”, Discrete Math. Appl., 20:3 (2010), 247–275 | DOI | DOI | MR | Zbl | Zbl

[22] Yakymiv A. L., “Asymptotics of the moments of the number of cycles of a random $A$-permutation”, Math. Notes, 88:5–6 (2010), 759–766 | DOI | DOI | MR | Zbl

[23] Yakymiv A. L., “On the order of random permutation with cycle weights”, Theory Probab. Appl., 63:2 (2018), 209–226 | DOI | DOI | MR | Zbl