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 Teor. Veroyatnost. i Primenen., 2003, Volume 48, Issue 4, Pages 818–828 (Mi tvp260)

Short Communications

Random mappings of finite sets with a known number of components

A. N. Timashev

Academy of Federal Security Service of Russian Federation

Abstract: We consider the class of all one-to-one mappings of an $n$-element set into itself, each of which has exactly $N$ connected components. Letting $n,N\to\infty$, we find that the asymptotic behavior of the mean and variance of the random variable is equal to the number of components of a given size in a mapping that is selected at random and is equiprobable among the elements of the mentioned class, and we prove the Poisson and local normal limit theorems for this random variable. Asymptotic estimates are found for the number of mappings with $N$ components, among which there are exactly $k$ components of a fixed size.

Keywords: random mapping, local limit theorem, asymptotic estimators, components.

DOI: https://doi.org/10.4213/tvp260

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English version:
Theory of Probability and its Applications, 2004, 48:4, 741–751

Bibliographic databases:

Citation: A. N. Timashev, “Random mappings of finite sets with a known number of components”, Teor. Veroyatnost. i Primenen., 48:4 (2003), 818–828; Theory Probab. Appl., 48:4 (2004), 741–751

Citation in format AMSBIB
\Bibitem{Tim03} \by A.~N.~Timashev \paper Random mappings of finite sets with a known number of components \jour Teor. Veroyatnost. i Primenen. \yr 2003 \vol 48 \issue 4 \pages 818--828 \mathnet{http://mi.mathnet.ru/tvp260} \crossref{https://doi.org/10.4213/tvp260} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2142528} \zmath{https://zbmath.org/?q=an:1060.60006} \transl \jour Theory Probab. Appl. \yr 2004 \vol 48 \issue 4 \pages 741--751 \crossref{https://doi.org/10.1137/S0040585X97980798} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000226305500014} 

• http://mi.mathnet.ru/eng/tvp260
• https://doi.org/10.4213/tvp260
• http://mi.mathnet.ru/eng/tvp/v48/i4/p818

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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. N. Timashov, “Limit theorems for the joint distribution of component sizes of a random mapping with a known number of components”, Discrete Math. Appl., 21:1 (2011), 39–46
2. A. N. Timashov, “Asymptotic expansions for the distribution of the number of components in random mappings and partitions”, Discrete Math. Appl., 21:3 (2011), 291–301
3. A. L. Yakymiv, “On the number of cyclic points of random $A$-mapping”, Discrete Math. Appl., 23:5-6 (2013), 503–515
4. A. L. Yakymiv, “On a number of components in a random $A$-mapping”, Theory Probab. Appl., 59:1 (2015), 114–127
5. A. L. Yakymiv, “On the Number of Components of Fixed Size in a Random $A$-Mapping”, Math. Notes, 97:3 (2015), 468–475
6. A. L. Yakymiv, “Limit theorems for the logarithm of the order of a random $A$-mapping”, Discrete Math. Appl., 27:5 (2017), 325–338
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