On the Number of $A$-Mappings

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 $n\to\infty$.