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 Diskr. Mat., 2001, Volume 13, Issue 4, Pages 60–72 (Mi dm310)

On the distribution of the number of cycles of a given length in the class of permutations with known number of cycles

A. N. Timashev

Abstract: We consider the set of all permutations of degree $n$ with $N$ cycles. We assume that the uniform distribution is defined on this set and consider the random variable equal to the number of cycles of a given length in the random permutation from this set. We obtain the asymptotic values of the mathematical expectation and the variance of this random variable and prove the limit theorems on the convergence to the Poisson and the Gaussian distributions as $n,N\to\infty$. We give the asymptotic expansions for the number of permutations of degree $n$ with $N$ cycles among which there are exactly $k=k(n,N)$ of a given length.

DOI: https://doi.org/10.4213/dm310

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English version:
Discrete Mathematics and Applications, 2001, 11:5, 471–483

Bibliographic databases:

UDC: 519.2

Citation: A. N. Timashev, “On the distribution of the number of cycles of a given length in the class of permutations with known number of cycles”, Diskr. Mat., 13:4 (2001), 60–72; Discrete Math. Appl., 11:5 (2001), 471–483

Citation in format AMSBIB
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\by A.~N.~Timashev
\paper On the distribution of the number of cycles of a given length in the class of
permutations with known number of cycles
\jour Diskr. Mat.
\yr 2001
\vol 13
\issue 4
\pages 60--72
\mathnet{http://mi.mathnet.ru/dm310}
\crossref{https://doi.org/10.4213/dm310}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1901783}
\zmath{https://zbmath.org/?q=an:1046.60008}
\transl
\jour Discrete Math. Appl.
\yr 2001
\vol 11
\issue 5
\pages 471--483

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• https://doi.org/10.4213/dm310
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This publication is cited in the following articles:
1. E. V. Cherepanova, “Limit distributions of the number of cycles of given length in a random permutation with a known number of cycles”, Discrete Math. Appl., 13:5 (2003), 507–522
2. A. V. Kolchin, “On limit theorems for the generalised allocation scheme”, Discrete Math. Appl., 13:6 (2003), 627–636
3. E. V. Cherepanova, “On the rate of convergence of the distribution of the number of cycles of given length in a random permutation with known number of cycles to the limit distributions”, Discrete Math. Appl., 16:4 (2006), 385–400
4. A. V. Kolchin, V. F. Kolchin, “On transition of distributions of sums of independent identically distributed random variables from one lattice to another in the generalised allocation scheme”, Discrete Math. Appl., 16:6 (2006), 527–540
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