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## Abstract

In this article, a random permutation *τ _{n}* is considered which is uniformly distributed on the set of all permutations of degree

*n*whose cycle lengths lie in a fixed set

*A*(the so-called

*A*-permutations). It is assumed that the set

*A*has an asymptotic density

*σ*> 0, and |

*k*:

*k*≤

*n*,

*k*∈

*A*,

*m*–

*k*∈

*A*|/

*n*→

*σ*

^{2}as

*n*→ ∞ uniformly in

*m*∈ [

*n*,

*Cn*] for an arbitrary constant

*C*> 1. The minimum degree of a permutation such that it becomes equal to the identity permutation is called the order of permutation. Let

*Z*be the order of a random permutation

_{n}*τ*. In this article, it is shown that the random variable ln

_{n}*Z*is asymptotically normal with mean

_{n}*l*(

*n*) = ∑

_{k∈A(n)}ln(

*k*)/

*k*and variance

*σ*ln

^{3}(

*n*)/3, where

*A*(

*n*) = {

*k*:

*k*∈

*A*,

*k*≤

*n*}. This result generalises the well-known theorem of P. Erdős and P. Turán where the uniform distribution on the whole symmetric group of permutations

*S*is considered, i.e., where

_{n}*A*is equal to the set of positive integers

**N**.

## 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 → ∞.

Let *S*
_{n} be the symmetric group of all permutations of degree *n*, *A* be some subset of the set of natural numbers ** N**, and

*T*

_{n}=

*T*

_{n}(

*A*) be the set of all permutations of

*S*

_{n}with cycle lengths belonging to

*A*. The permutations of

*T*

_{n}are called

*A*-permutations. We consider a wide class of the sets

*A*with the asymptotic density

*σ*> 0. In this article, the limit distributions are obtained for

*μ*

_{m}(

*n*)/

*n*as

*n*→ ∞ and

*m*∈

**is fixed. Here**

*N**μ*(

_{m}*n*) is the length of the

*m*th maximal cycle in a random permutation uniformly distributed on

*T*

_{n}. It is shown here that these limit distributions coincide with the limit distributions of the corresponding functionals of the random permutations in the Ewens model with parameter σ.