Asymptotic properties of matrices related to mappings of partitions

Let $S=(S_1,\dots,S_{\tau})$ be a partition of the set $\mathscr{N}=\{1,\dots,n\}$ into nonempty disjoint subsets, $\Phi$ a permutation on $\mathscr{N}$, and $\xi_{ij}=|\Phi S_i\cap S_j|$ the cardinality of the intersection of the sets $\Phi S_i$ and $S_j$. Assuming that $S$ is selected at random and equiprobably from the set of all the permutations satisfying the condition $|S_i|=s_i$, $i=1\ldots r$, and the permutation $\Phi$ (possibly random) satisfies some constrains, local and integral limit theorems are proved for the joint distribution of the random variables $\xi_{ij}$, $i,j=1\ldots r$, as $n\to\infty$ and $s_in^{-1}\to a_j\in(0,1)$.