On the Number of Components of Fixed Size in a Random $A$-Mapping

Let $\mathfrak S_n$ be the semigroup of mappings of a set of $n$ elements into itself, let $A$ be a fixed subset of the set of natural numbers $\mathbb N$, and let $V_n(A)$ be the set of mappings from $\mathfrak S_n$ for which the sizes of the contours belong to the set $A$. Mappings from $V_n(A)$ are usually called $A$-mappings. Consider a random mapping $\sigma_n$ uniformly distributed on $V_n(A)$. It is assumed that the set $A$ possesses asymptotic density $\varrho$, including the case $\varrho=0$. Let $\xi_{in}$ be the number of connected components of a random mapping $\sigma_n$ of size $i\in\mathbb N$. For a fixed integer $b\in\mathbb N$, as $n\to\infty$, the asymptotic behavior of the joint distribution of random variables $\xi_{1n},\xi_{2n},\dots,\xi_{bn}$ is studied. It is shown that this distribution weakly converges to the joint distribution of independent Poisson random variables $\eta_1,\eta_2,\dots,\eta_b$ with some parameters $\lambda_i=\mathsf E\eta_{i}$, $i\in\mathbb N$.