On a Number of Components in a Random $A$-Mapping
Arsen Lubomirovich Yakymiv · Theory of Probability and Its Applications · 2015
Suppose that ${\frak S}_n$ is the semigroup of all mappings of the set of $n$ elements into itself, $A$ is a fixed subset of the set of natural numbers ${\bf N}$, and $V_n(A)$ is the set of mappings from ${\frak S}_n$ whose contours are of sizes belonging to $A$. Mappings from $V_n(A)$ are usually called $A$-mappings. Consider a random mapping $\sigma_n$, uniformly distributed on $V_n(A)$. It is assumed that the set $A$ has an asymptotic density $\varrho>0$. Let $ u_n$ be a number of connected components of a random mapping $\sigma_n$. In the present paper, it is shown that a random variable $ u_n$ is asymptotically normal with mathematical expectation $a(n)=\sum_{k\in A(\sqrt{n})}1/{k}$ and variance $\varrho\log(n)/2$, where $A(t)=\{k\!:$ $k\in A,\ k\le t\}$.