The cyclic structure of random permutations

  • V. F. Kolchin
  • V. P. Chistyakov


Letαr denote the number of cycles of length r in a random permutation, taking its values with equal probability from among the set Sn of all permutations of length n. In this paper we study the limiting behavior of linear combinations of random permutationsα1, ...,αr having the form
$$\zeta _{n, r} = c_{r1^{a_1 } } + ... + c_{rr} a_r $$
in the case when n, r→∞. We shall show that the class of limit distributions forξn,r as n, r→∞ and r In r/h→0 coincides with the class of unbounded divisible distributions. For the random variables ηn, r=α1+2α2+... rαr, equal to the number of elements in the permutation contained in cycles of length not exceeding r, we find' limit distributions of the form r In r/n→0 and r=γn, 0<γ<1.


Linear Combination Equal Probability Limit Distribution Random Permutation Cyclic Structure 
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Literature cited

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    V. A. Goncharov, “From the domain of combinatorics,” Izv. Akad. Nauk SSSR, Ser. Matem.,8, No. 1, 3–48 (1944).Google Scholar
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    B. V. Gnedenko and A. N. Kolmogorov, Limit Distributions for Sums of Independent Random Variables [in Russian], Gostekhizdat, Moscow-Leningrad (1949).Google Scholar
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    V. F. Kolchin, “A problem on the distribution of particles among cells, and cycles of random permutations,” Teoriya Veroyatnostei i Primen.,16, No. 1, 67–82 (1971).Google Scholar

Copyright information

© Plenum Publishing Corporation 1976

Authors and Affiliations

  • V. F. Kolchin
    • 1
  • V. P. Chistyakov
    • 1
  1. 1.V. A. Steklov Mathematics InstituteAcademy of Sciences of the USSRUSSR

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