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On the number of $$A$$-mappings. (English. Russian original) Zbl 1176.60005
Math. Notes 86, No. 1, 132-139 (2009); translation from Mat. Zametki 86, No. 1, 139-147 (2009).
Summary: Suppose that $$\mathfrak{S}_n$$ is the semigroup of mappings of the set of $$n$$ elements into itself, $$A$$ is a fixed subset of the set of natural numbers $$\mathbb N$$, and $$V_n (A)$$ is the set of mappings from $$\mathfrak{S}_n$$ 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 $$V_n(A)$$. Suppose that $$\nu _n$$ is the number of components and $$\lambda _n$$ is the number of cyclic points of the random mapping $$\sigma _n$$ . In this paper, for a particular class of sets $$A$$, we obtain the asymptotics of the number of elements of the set $$V_n (A)$$ and prove limit theorems for the random variables $$\nu _n$$ and $$\lambda _n$$ as $$n \rightarrow \infty$$.

##### MSC:
 60C05 Combinatorial probability 60F05 Central limit and other weak theorems
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