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Moment Characteristics of a Random Mapping with Restrictions on Component Sizes

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Abstract

Let \(\mathfrak{S}_n\) be the semigroup of mappings of an \(n\)-element set \(X\) into itself. For a set \(D\subseteq\mathbb N\), denote by \(\mathfrak{S}_n(D)\) the family of those mappings in \(\mathfrak{S}_n\) whose component sizes belong to \(D\). Suppose that a random mapping \(\sigma_n=\sigma_n(D)\) is uniformly distributed on \(\mathfrak{S}_n(D)\). We consider a class of sets \(D\subseteq\mathbb N\) with positive densities in the set \(\mathbb N\) of positive integers. Let \(\zeta_n\) be the number of components of the random mapping \(\sigma_n\). We find asymptotic formulas for the expectation and variance of the random variable \(\zeta_n\) as \(n\to\infty\).

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Acknowledgments

I am deeply grateful to the referee for pointing out some errors and making other valuable comments.

Funding

This work is supported by the Russian Science Foundation under grant 19-11-00111.

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Authors and Affiliations

  1. Steklov Mathematical Institute of Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991, Russia

    A. L. Yakymiv

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  1. A. L. Yakymiv
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Correspondence to A. L. Yakymiv.

Additional information

Translated from Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2022, Vol. 316, pp. 376–389 https://doi.org/10.4213/tm4214.

Translated by I. Nikitin

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Yakymiv, A.L. Moment Characteristics of a Random Mapping with Restrictions on Component Sizes. Proc. Steklov Inst. Math. 316, 356–369 (2022). https://doi.org/10.1134/S0081543822010242

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