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 Diskr. Mat., 2000, Volume 12, Issue 4, Pages 53–62 (Mi dm357)

On permutations with cycle lengths from a random set

A. L. Yakymiv

Abstract: Let $\xi_1,…,\xi_n,…$ be a sequence of independent Bernoulli random variables which take the value 1 with probability $\sigma\in (0,1]$. Given this sequence, we construct the random set $A\subseteq\mathbf N=\{1,2,3,…\}$ as follows: a number $n\in\mathbf N$ is included in $A$ if and only if $\xi_n=1$. Let $T_n=T_n(A)$ denote the set of the permutations of degree $n$ whose cycle lengths belong to the set $A$. In this paper, we find the asymptotic behaviour of the number of elements of the set $T_n(A)$ as $n\to\infty$.
For any fixed $A$, the uniform distribution is defined on $T_n(A)$. Under these hypotheses, limit theorems are obtained for the total number of cycles and the number of cycles of a fixed length in a random permutation in $T_n(A)$.
Similar problems were earlier solved for various classes of deterministic sets $A$.
This research was supported by the Russian Foundation for Basic Research, grants 00–15–96136 and 00–01–00090.

DOI: https://doi.org/10.4213/dm357

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English version:
Discrete Mathematics and Applications, 2000, 10:6, 543–551

Bibliographic databases:

UDC: 519.2

Citation: A. L. Yakymiv, “On permutations with cycle lengths from a random set”, Diskr. Mat., 12:4 (2000), 53–62; Discrete Math. Appl., 10:6 (2000), 543–551

Citation in format AMSBIB
\Bibitem{Yak00} \by A.~L.~Yakymiv \paper On permutations with cycle lengths from a random set \jour Diskr. Mat. \yr 2000 \vol 12 \issue 4 \pages 53--62 \mathnet{http://mi.mathnet.ru/dm357} \crossref{https://doi.org/10.4213/dm357} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1826179} \zmath{https://zbmath.org/?q=an:1046.60009} \transl \jour Discrete Math. Appl. \yr 2000 \vol 10 \issue 6 \pages 543--551 

• http://mi.mathnet.ru/eng/dm357
• https://doi.org/10.4213/dm357
• http://mi.mathnet.ru/eng/dm/v12/i4/p53

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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. A. L. Yakymiv, “On the distribution of the $m$th maximal cycle lengths of random $A$-permutations”, Discrete Math. Appl., 15:5 (2005), 527–546
2. A. L. Yakymiv, “Limit theorem for the general number of cycles in a random $A$-permutation”, Theory Probab. Appl., 52:1 (2008), 133–146
3. Lugo, M, “Profiles of permutaions”, Electronic Journal of Combinatorics, 16:1 (2009), R99
4. A. L. Yakymiv, “Asymptotics of the Moments of the Number of Cycles of a Random $A$-Permutation”, Math. Notes, 88:5 (2010), 759–766
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