RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PERSONAL OFFICE
 General information Latest issue Archive Impact factor Subscription Guidelines for authors License agreement Submit a manuscript Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Mat. Zametki: Year: Volume: Issue: Page: Find

 Mat. Zametki, 1975, Volume 18, Issue 6, Pages 929–938 (Mi mz7716)

The cyclic structure of random permutations

V. F. Kolchin, V. P. Chistyakov

V. A. Steklov Mathematical Institute, USSR Academy of Sciences

Abstract: Let $\alpha_r$ denote the number of cycles of length $r$ in a random permutation, taking its values with equal probability from among the set $S_n$ of all permutations of length $n$. In this paper we study the limiting behavior of linear combinations of random permutations $\alpha_1,…,\alpha_r$ having the form
$$\zeta_{n,r}=C_{r1}\alpha_1+…+C_{rr}\alpha_r$$
in the case when $n,r\to\infty$. We shall show that the class of limit distributions for $\zeta_{n,r}$ as $n,r\to\infty$ and $r\ln r/n\to0$ coincides with the class of unbounded divisible distributions. For the random variables $\eta_{n,r}=\alpha_1+2\alpha_2+…+r\alpha_r$, equal to the number of elements in the permutation contained in cycles of length not exceeding $r$, we find limit distributions of the form $r\ln r/n\to0$ è $r=\gamma n$, $0<\gamma<1$.

Full text: PDF file (508 kB)

English version:
Mathematical Notes, 1975, 18:6, 1139–1144

Bibliographic databases:

Document Type: Article
UDC: 519.2

Citation: V. F. Kolchin, V. P. Chistyakov, “The cyclic structure of random permutations”, Mat. Zametki, 18:6 (1975), 929–938; Math. Notes, 18:6 (1975), 1139–1144

Citation in format AMSBIB
\Bibitem{KolChi75} \by V.~F.~Kolchin, V.~P.~Chistyakov \paper The cyclic structure of random permutations \jour Mat. Zametki \yr 1975 \vol 18 \issue 6 \pages 929--938 \mathnet{http://mi.mathnet.ru/mz7716} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=402852} \zmath{https://zbmath.org/?q=an:0344.60015} \transl \jour Math. Notes \yr 1975 \vol 18 \issue 6 \pages 1139--1144 \crossref{https://doi.org/10.1007/BF01099997} 

• http://mi.mathnet.ru/eng/mz7716
• http://mi.mathnet.ru/eng/mz/v18/i6/p929

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Yu. V. Bolotnikov, V. N. Sachkov, V. E. Tarakanov, “On some classes of random variables on cycles of permutations”, Math. USSR-Sb., 36:1 (1980), 87–99
2. A. N. Timashev, “On the distribution of the number of cycles of a given length in the class of permutations with known number of cycles”, Discrete Math. Appl., 11:5 (2001), 471–483
•  Number of views: This page: 153 Full text: 46 First page: 1