Mathematical Notes

, Volume 88, Issue 5–6, pp 759–766 | Cite as

Asymptotics of the moments of the number of cycles of a random A-permutation

Article
  • 48 Downloads

Abstract

We consider random permutations uniformly distributed on the set of all permutations of degree n whose cycle lengths belong to a fixed set A (the so-called A-permutations). In the present paper, we establish an asymptotics of the moments of the total number of cycles and of the number of cycles of given length of this random permutation as n → ∞.

Keywords

random A-permutation number of cycles of a permutation uniform distribution moments of the total number of cycles slowly varying function 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    V. N. Sachkov, Probabilistic Methods in Combinatorial Analysis in Encyclopedia of Mathematics and Its Applications (Cambridge University Press, Cambridge, 1997; Nauka, Moscow, 1978) Vol. 56.Google Scholar
  2. 2.
    E. A. Bender, “Asymptotic methods in enumeration,” SIAM Rev. 16(4), 485–515 (1974).MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Yu. V. Bolotnikov, V. N. Sachkov, and V. E. Tarakanov, “Asymptotic normality of some quantities related to the cyclic structure of random permutations,” Mat. Sb. 99(1), 121–133 (1976).MathSciNetGoogle Scholar
  4. 4.
    M. P. Mineev and A. I. Pavlov, “On the number of permutations of special form,” Mat. Sb. 99(3), 468–476 (1976) [Math. USSR-Sb. 28 (3), 421–429 (1978)].MathSciNetGoogle Scholar
  5. 5.
    M. P. Mineev and A. I. Pavlov, “On an equation in permutations,” in Trudy Mat. Inst. Steklov, Vol. 142: Number Theory, Mathematical Analysis, and Their Applications, Collection of papers dedicated to Academician I. M. Vinogradov on the occasion of his 85th birthday (Nauka, Moscow, 1976), pp. 182–194, [Proc. Steklov Inst.Math. 142, 195–208 (1979)].Google Scholar
  6. 6.
    V. N. Sachkov, “Mappings of a finite set with limitations on contours and height,” Teor. Veroyatnost. Primenen. 17(4), 679–694 (1972) [Theory Probab. Appl. 17, 640–656 (1972)].MathSciNetGoogle Scholar
  7. 7.
    V. N. Sachkov, “Random mappings with bounded height,” Teor. Veroyatnost. Primenen. 18(1), 122–132 (1973) [Theory Probab. Appl. 18, 120–130 (1973)].Google Scholar
  8. 8.
    V. F. Kolchin, Random Mappings (Nauka, Moscow, 1984; Optimization Software, Inc., Publications Division, New York, 1986).Google Scholar
  9. 9.
    V. F. Kolchin, Random Graphs (Cambridge University Press, Cambridge, 1999; Fizmatlit, Moscow, 2000).MATHGoogle Scholar
  10. 10.
    V. N. Sachkov, Combinatorial Methods of Discrete Mathematics (Nauka, Moscow, 1977; Cambridge University Press, Cambridge, 1996).Google Scholar
  11. 11.
    A. L. Yakymiv, Probabilistic Applications of Tauberian Theorems (Fizmatlit, Moscow, 2005; VSP, Leiden, 2005).MATHGoogle Scholar
  12. 12.
    A. L. Yakymiv, “On the distribution of the mth maximal cycle lengths of random A-permutations,” Diskretn. Mat. 17(4), 40–58 (2005) [DiscreteMath. Appl. 15 (5), 527–546 (2005)].MathSciNetGoogle Scholar
  13. 13.
    V. L. Goncharov, “From the domain of combinatorics,” Izv. Akad. Nauk SSSR Ser. Mat. 8(1), 3–48 (1944).MATHGoogle Scholar
  14. 14.
    A. N. Timashev, “Limit theorems for allocation of particles over different cells with restrictions to the size of the cells,” Teor. Veroyatnost. Primenen. 49(4), 712–725 (2004) [Theory Probab. Appl. 49 (4), 659–670 (2004)].MathSciNetGoogle Scholar
  15. 15.
    E. Seneta, Regularly Varying Functions (Springer-Verlag, Berlin-Heidelberg-New York, 1976; Nauka, Moscow, 1985).MATHCrossRefGoogle Scholar
  16. 16.
    L. M. Volynets, “An example of nonstandard asymptotics of the number of permutations with constraints on the lengths of the cycles,” in Probabilistic processes and their applications (Moskov. Inst. Elektron. Mashinostroeniya, Moscow, 1989, Moscow, 1989), pp. 85–90 [in Russian].Google Scholar
  17. 17.
    V. F. Kolchin, “On the number of permutations with constraints on their cycle lengths,” Diskretn. Mat. 1(2), 97–109 (1989) [Discrete Math. Appl. 1 (2), 179–193 (1991)].MathSciNetGoogle Scholar
  18. 18.
    V. F. Kolchin, “The number of permutations with cycle lengths from a fixed set,” in Random Graphs, Poznan’, 1989 (Wiley, New York, 1992), Vol. 2, pp. 139–149.Google Scholar
  19. 19.
    A. I. Pavlov, “Some classes of permutations with number-theoretic constraints on the lengths of the cycles,” Mat. Sb. 129(2), 252–263 (1986) [Math. USSR-Sb. 57 (1), 263–275 (1987)].MathSciNetGoogle Scholar
  20. 20.
    A. I. Pavlov, “On permutations with cycle lengths from a given set,” in Abstracts of Papers Delivered at the Seminar “Probabilistic Methods in Discrete Mathematics (April-December, 1985)” Teor. Veroyatnost. Primenen. 31(3), 617–624 (1986) [Theory Probab. Appl. 31 (3), 544–550 (1986)].Google Scholar
  21. 21.
    A. I. Pavlov, “On the number of substitutions with cycle lengths from a given set,” Diskretn. Mat. 3(3), 109–123 (1991) [Discrete Math. Appl. 2 (4), 445–459 (1992)].MATHGoogle Scholar
  22. 22.
    A. I. Pavlov, “Asymptotics of the number of permutations with number-theoretic restrictions on cycle length,” Dokl. Ross. Akad. Nauk 335(5), 556–559 (1994) [Russian Acad. Sci. Dokl. Math. 49 (2), 380–385 (1994)].Google Scholar
  23. 23.
    A. I. Pavlov, “On two classes of permutations with number-theoretic conditions on the lengths of the cycles,” Mat. Zametki 62(6), 881–891 (1997) [Math. Notes 62 (5–6), 739–746 (1997)].Google Scholar
  24. 24.
    A. L. Yakymiv, “On the number of A-permutations,” Mat. Sb. 180(2), 294–303 (1989) [Math. USSR-Sb. 66 (1), 301–311 (1990)].Google Scholar
  25. 25.
    A. L. Yakymiv, “Permutations with cycle lengths in a given set.,” Diskretn. Mat. 1(1), 125–134 (1989) [DiscreteMath. Appl. 1 (1), 105–116 (1991)].MathSciNetGoogle Scholar
  26. 26.
    A. L. Yakymiv, “Some classes of permutations with cycle lengths in a given set,” Diskretn. Mat. 4(3), 128–134 (1992) [Discrete Math. Appl. 3 (2), 213–220 (1993)].Google Scholar
  27. 27.
    A. L. Yakymiv, “Limit theorems for random A-permutations,” in Progr. Pure Appl. Discrete Math., Vol. 1: Probabilistic Methods in Discrete Mathematics (Petrozavodsk, 1992; VSP, Utrecht, 1993), pp. 459–469.Google Scholar
  28. 28.
    A. L. Yakymiv, “On permutations with cycle lengths in a random set,” Diskretn. Mat. 12(4), 53–62 (2000) [Discrete Math. Appl. 10 (6), 543–551 (2000)].MathSciNetGoogle Scholar
  29. 29.
    I. S. Gradshtein and I. M. Ryzhik, Tables of Integrals, Sums, Series, and Products (Fizmatgiz, Moscow, 1962; Academic Press, New York-London, 1965).Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2010

Authors and Affiliations

  1. 1.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia

Personalised recommendations