On the distribution of the mth maximal cycle lengths of random A-permutations
Arsen Lubomirovich Yakymiv · Discrete Mathematics and Applications · 2005
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 ∈ N is fixed. Here 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.