Random Mappings and Decompositions of Finite Sets

Let $X=\{1,2,\dots,n\}$ be a finite set,
\begin{equation} X=S_1+\cdots+S_r \end{equation}
be a partition of $X$.
\begin{equation} \Phi=\begin{pmatrix} 1 & 2 & \dots & n\\ \varphi_1 & \varphi_2 & \ldots & \varphi_n\\ \end{pmatrix} \end{equation}
be a permutation of elements of $X$, $N(A)$ be the number of elements of any finite set $A$. We denote by $R(s_1,\dots,s_r)$ the set of all partitions (1) with $N(S_j)=s_j$, $j=1,\dots,r$, and by $T(z_1,\dots,z_m)$ the set of all permutations (2) with cycles of lengths $z_1\le z_2\le\dots\le z_m$.