Abstract
Let S n be the semigroup of mappings of the set of n elements into itself, A be a fixed subset of the set of natural numbers ℕ, and V n (A) be the set of mappings from S n with cycle lengths belonging to A. Mappings from V n (A) are called A-mappings. Consider a random mapping σ n uniformly distributed on V n (A). Let λ n be the number of cyclic points of σ n . It is supposed that the set A has an asymptotic nonnegative density ς. We describe the asymptotic behaviour of the cardinality of the set V n (A) and prove a limit theorem for the sequence of random variables λ n as n → ∞.
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