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On the distribution of the $$m$$th maximal cycle lengths of random $$A$$-permutations. (English, Russian) Zbl 1101.60004
Discrete Math. Appl. 15, No. 5, 527-546 (2005); translation from Diskretn. Mat. 17, No. 4, 40-58 (2005).
Summary: Let $$S_n$$ be the symmetric group of all permutations of degree $$n$$, $$A$$ be some subset of the set of natural numbers $${\mathbf N}$$, 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 $$\sigma>0$$. The limit distributions are obtained for $$\mu_m(n)/n$$ as $$n\to\infty$$ and $$m\in{\mathbf N}$$ is fixed. Here $$\mu_m(n)$$ 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 $$\sigma$$.

##### MSC:
 60C05 Combinatorial probability 60F05 Central limit and other weak theorems
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