Size distribution of the largest component of a random $A$-mapping

Let $\mathfrak S_n$ be a semigroup of all mappings from the $n$-element set $X$ into itself. We consider a set $\mathfrak S_n(A)$ of mappings from $\mathfrak S_n$ such that their contour sizes belong to the set $A\subseteq N$. These mappings are called $A$-mappings. Let a random mapping $\tau_n$ have a distribution on $\mathfrak S_n(A)$ such that each connected component with volume $i\in N$ have weight $\vartheta_i\geq 0$. Let $D$ be a subset of $N$. It is assumed that $\vartheta_i\to\vartheta>0$ for $i\in D$ and $\vartheta_i\to0$ for $i\in N\setminus D$ as $i\to\infty$. Let $\mu(n)$ be the maximal volume of components of the random mapping $\tau_n$ . We suppose that sets $A$ and $D$ have asymptotic densities $\varrho>0$ and $\rho>0$ in $N$ respectively. It is shown that the random variables $\mu(n)/n$ converge weakly to random variable $\nu$ as $n\to\infty$. The distribution of $\nu$ coincides with the limit distribution of the corresponding characteristic in the Ewens sampling formula for random permutation with the parameter $\rho\varrho\vartheta/2$.