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• Author: A. L. YAKYMIV
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Abstract

In this article, a random permutation τn 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: kn, kA, mkA|/nσ 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 Zn be the order of a random permutation τn. In this article, it is shown that the random variable ln Zn is asymptotically normal with mean l(n) = ∑kA(n) ln(k)/k and variance σ ln3(n)/3, where A(n) = {k: kA, kn}. This result generalises the well-known theorem of P. Erdős and P. Turán where the uniform distribution on the whole symmetric group of permutations Sn is considered, i.e., where A is equal to the set of positive integers N.

Abstract

Let Sn 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 Vn(A) be the set of mappings from Sn with cycle lengths belonging to A. Mappings from Vn(A) are called A-mappings. Consider a random mapping σn uniformly distributed on Vn(A). Let λn be the number of cyclic points of σn. It is supposed that the set A has an asymptotic nonnegative density ς. We describe the asymptotic behaviour of the cardinality of the set Vn(A) and prove a limit theorem for the sequence of random variables λn 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 mN is fixed. Here μm(n) is the length of the mth 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.