Some classes of random mappings of finite sets, and nonhomogeneous branching processes

Let $X=\bigcup_{t=0}^TX_t$ be a finite set, where $X_t$, $t=1,2,\ldots,T$, are pairwise non-overlapping sets, $N_t=|X_t|$ be the cardinality of the set $X_t$, $t=0,1,\ldots,T$. Let $\mathcal F_1$ be the class of all mappings $f$ of the set $X'=X\setminus X_0$ into $X$ such that the image $y=f(x)\in X_{t-1}\cup X_t$ for any $x\in X_t$, $t=1,\ldots,T$. The cardinality of the set of all mappings of the class $\mathcal F_1$ is $\prod_{t=1}^T(N_{t-1}+N_t)^{N_t}$. With the use of non-homogeneous branching processes, we study some asymptotical properties of the uniformly distributed on $\mathcal F_1$ random mapping $f$ as $N_t\to\infty$, $t=1,2,\ldots,T$. Similar results are obtained for some other classes of random mappings $f$ of the set $X$.
This research was supported by the Russian Foundation for Basic Research, grant 02.01.00266, and the grant 1758.2003.1 of the President of Russian Federation for support of leading scientific schools.