On random mappings with restrictions on component sizes

Let $\mathfrak{S}_{n}$ be a semigroup of mappings from the $n$-element set $X$ into itself and let $\mathfrak{S}_{n}(A)$ be a set of mappings from $\mathfrak{S}_{n}$ whose component sizes belong to the set $A$. By $\sigma_n=\sigma_n(A)$ we denote a random mapping having a uniform distribution on the set $\mathfrak{S}_{n}(A)$. Such objects were considered by A.N. Timashev in 2019. For a certain class of sets $A$ having positive densities in the set $N$ of natural numbers, the asymptotic number of elements of the set $\mathfrak{S}_{n}(A)$ is found for $n\rightarrow\infty$. An estimate is also obtained for the total variation distance between the $\sigma_n(A)$ mapping structure and the corresponding sequence of independent Poisson random variables.