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 Mat. Zametki, 2009, Volume 86, Issue 1, Pages 139–147 (Mi mz4521)

On the Number of $A$-Mappings

A. L. Yakymiv

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: Suppose that $\mathfrak S_n$ is the semigroup of mappings of the set of $n$ elements into itself, $A$ is a fixed subset of the set of natural numbers $\mathbb N$, and $V_n(A)$ is the set of mappings from $\mathfrak S_n$ whose contours are of sizes belonging to $A$. Mappings from $V_n(A)$ are usually called $A$-mappings. Consider a random mapping $\sigma_n$, uniformly distributed on $V_n(A)$. Suppose that $\nu_n$ is the number of components and $\lambda_n$ is the number of cyclic points of the random mapping $\sigma_n$. In this paper, for a particular class of sets $A$, we obtain the asymptotics of the number of elements of the set $V_n(A)$ and prove limit theorems for the random variables $\nu_n$ and $\lambda_n$ as $n\to\infty$.

Keywords: $A$-mapping, symmetric semigroup of mappings, random mapping, random variable, Euler gamma function, uniform distribution

DOI: https://doi.org/10.4213/mzm4521

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English version:
Mathematical Notes, 2009, 86:1, 132–139

Bibliographic databases:

UDC: 519.2
Revised: 26.11.2008

Citation: A. L. Yakymiv, “On the Number of $A$-Mappings”, Mat. Zametki, 86:1 (2009), 139–147; Math. Notes, 86:1 (2009), 132–139

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/mz4521
• https://doi.org/10.4213/mzm4521
• http://mi.mathnet.ru/eng/mz/v86/i1/p139

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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 number of cyclic points of random $A$-mapping”, Discrete Math. Appl., 23:5-6 (2013), 503–515
2. A. L. Yakymiv, “On a number of components in a random $A$-mapping”, Theory Probab. Appl., 59:1 (2015), 114–127
3. 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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