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This article is cited in 2 scientific papers (total in 2 papers)
The cyclic structure of random permutations
V. F. Kolchin, V. P. Chistyakov V. A. Steklov Mathematical Institute, USSR Academy of Sciences
Abstract:
Let $\alpha_r$ denote the number of cycles of length $r$ in a random permutation, taking its values with equal probability from among the set $S_n$ of all permutations of length $n$. In this paper we study the limiting behavior of linear combinations of random permutations $\alpha_1,\dots,\alpha_r$ having the form
$$
\zeta_{n,r}=C_{r1}\alpha_1+\dots+C_{rr}\alpha_r
$$
in the case when $n,r\to\infty$. We shall show that the class of limit distributions for $\zeta_{n,r}$ as $n,r\to\infty$ and $r\ln r/n\to0$ coincides with the class of unbounded divisible distributions. For the random variables $\eta_{n,r}=\alpha_1+2\alpha_2+\dots+r\alpha_r$, equal to the number of elements in the permutation contained in cycles of length not exceeding $r$, we find limit distributions of the form $r\ln r/n\to0$ и $r=\gamma n$, $0<\gamma<1$.
Received: 11.11.1975
Citation:
V. F. Kolchin, V. P. Chistyakov, “The cyclic structure of random permutations”, Mat. Zametki, 18:6 (1975), 929–938; Math. Notes, 18:6 (1975), 1139–1144
Linking options:
https://www.mathnet.ru/eng/mzm7716 https://www.mathnet.ru/eng/mzm/v18/i6/p929
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