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 Diskr. Mat., 2005, Volume 17, Issue 4, Pages 40–58 (Mi dm128)

On the distribution of the $m$th maximal cycle lengths of random $A$-permutations

A. L. Yakymiv

Abstract: Let $S_n$ be the symmetric group of all permutations of degree $n$, $A$ be some subset of the set of natural numbers $\mathbf N$, and $T_n=T_n(A)$ be the set of all permutations of $S_n$ with cycle lengths belonging to $A$. The permutations of $T_n$ are called $A$-permutations. We consider a wide class of the sets $A$ with the asymptotic density $\sigma>0$. In this article, the limit distributions are obtained for $\mu_{m}(n)/n$ as $n\to\infty$ and $m\in\mathbf N$ is fixed. Here $\mu_{m}(n)$ is the length of the $m$th maximal cycle in a random permutation uniformly distributed on $T_n$. It is shown here that these limit distributions coincide with the limit distributions of the corresponding functionals of the random permutations in the Ewens model with parameter $\sigma$.
This research was supported by the Russian Foundation for Basic Research, grant 05–01–00583, and by the Program of the President of the Russian Federation for support of leading scientific schools, grant 1758.2003.1.

DOI: https://doi.org/10.4213/dm128

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English version:
Discrete Mathematics and Applications, 2005, 15:5, 527–546

Bibliographic databases:

UDC: 519.2
Revised: 15.03.2005

Citation: A. L. Yakymiv, “On the distribution of the $m$th maximal cycle lengths of random $A$-permutations”, Diskr. Mat., 17:4 (2005), 40–58; Discrete Math. Appl., 15:5 (2005), 527–546

Citation in format AMSBIB
\Bibitem{Yak05} \by A.~L.~Yakymiv \paper On the distribution of the $m$th maximal cycle lengths of random $A$-permutations \jour Diskr. Mat. \yr 2005 \vol 17 \issue 4 \pages 40--58 \mathnet{http://mi.mathnet.ru/dm128} \crossref{https://doi.org/10.4213/dm128} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2240540} \zmath{https://zbmath.org/?q=an:1101.60004} \elib{https://elibrary.ru/item.asp?id=9154201} \transl \jour Discrete Math. Appl. \yr 2005 \vol 15 \issue 5 \pages 527--546 \crossref{https://doi.org/10.1515/156939205776368931} 

• http://mi.mathnet.ru/eng/dm128
• https://doi.org/10.4213/dm128
• http://mi.mathnet.ru/eng/dm/v17/i4/p40

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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, “Random $A$-Permutations: Convergence to a Poisson Process”, Math. Notes, 81:6 (2007), 840–846
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. A. L. Yakymiv, “On the Number of $A$-Mappings”, Math. Notes, 86:1 (2009), 132–139
4. A. L. Yakymiv, “Limit Theorem for the Middle Members of Ordered Cycle Lengths in Random $A$-Permutations”, Theory Probab. Appl., 54:1 (2010), 114–128
5. A. L. Yakymiv, “A limit theorem for the logarithm of the order of a random $A$-permutation”, Discrete Math. Appl., 20:3 (2010), 247–275
6. A. L. Yakymiv, “Asymptotics of the Moments of the Number of Cycles of a Random $A$-Permutation”, Math. Notes, 88:5 (2010), 759–766
7. Benaych-Georges F., “Cycles of free words in several independent random permutations with restricted cycle lengths”, Indiana Univ. Math. J., 59:5 (2010), 1547–1586
8. A. L. Yakymiv, “Random $A$-permutations and Brownian motion”, Proc. Steklov Inst. Math., 282 (2013), 298–318
9. A. L. Yakymiv, “On the order of random permutation with cycle weights”, Theory Probab. Appl., 63:2 (2018), 209–226
10. A. L. Yakymiv, “Asymptotics with remainder term for moments of the total cycle number of random $A$-permutation”, Discrete Math. Appl., 31:1 (2021), 51–60
11. A. L. Yakymiv, “Size distribution of the largest component of a random $A$-mapping”, Discrete Math. Appl., 31:2 (2021), 145–153
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