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→∞, 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.
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
\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
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\crossref{https://doi.org/10.4213/tvp260}
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\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{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000226305500014}
Linking options:
https://www.mathnet.ru/eng/tvp260
https://doi.org/10.4213/tvp260
https://www.mathnet.ru/eng/tvp/v48/i4/p818
This publication is cited in the following 6 articles:
A. L. Yakymiv, “Limit theorems for the logarithm of the order of a random A-mapping”, Discrete Math. Appl., 27:5 (2017), 325–338
A. L. Yakymiv, “On the Number of Components of Fixed Size in a Random A-Mapping”, Math. Notes, 97:3 (2015), 468–475
A. L. Yakymiv, “On a number of components in a random A-mapping”, Theory Probab. Appl., 59:1 (2015), 114–127
A. L. Yakymiv, “On the number of cyclic points of random A-mapping”, Discrete Math. Appl., 23:5-6 (2013), 503–515
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
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