A limit theorem for the logarithm of the order of a random A-permutation
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A. L. Yakymiv
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: k ≤ n, k ∈ A, m – k ∈ A|/n → σ2 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) = ∑k∈A(n) ln(k)/k and variance σ ln3(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 Sn is considered, i.e., where A is equal to the set of positive integers N.
© de Gruyter 2010
Articles in the same Issue
- A limit theorem for the logarithm of the order of a random A-permutation
- On game-theoretic characterisation of stochastic independence
- On the potential divisibility of matrices over distributive lattices
- On learning monotone Boolean functions with irrelevant variables
- Barriers of perfectly balanced Boolean functions
- On the classification of Post automaton bases by the decidability of the A-completeness property for definite automata
Articles in the same Issue
- A limit theorem for the logarithm of the order of a random A-permutation
- On game-theoretic characterisation of stochastic independence
- On the potential divisibility of matrices over distributive lattices
- On learning monotone Boolean functions with irrelevant variables
- Barriers of perfectly balanced Boolean functions
- On the classification of Post automaton bases by the decidability of the A-completeness property for definite automata
