Moment Characteristics of a Random Mapping with Restrictions on Component Sizes

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 $n\to \infty$.