Variance of the number of cycles of random A-permutation
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Arsen L. Yakymiv
Abstract
We consider random permutations having uniform distribution on the set of all permutations of the n-element set with lengths of cycles belonging to a fixed set A (so-called A-permutations). For some class of sets A the asymptotic formula for the variance of the number of cycles of such permutations is obtained.
Note: Originally published in Diskretnaya Matematika (2020) 32,№3, 135–146 (in Russian).
Acknowledgment
The author is grateful to the reviewer for valuable notes.
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Funding: This work was supported by the Russian Science Foundation under grant 19-11-00111.
References
[1] Billingsley P., Convergence of Probability Measures, New York: John Wiley & Sons, 1968, xii+253 pp.Search in Google Scholar
[2] Bolotnikov Yu. V., Sachkov V. N., Tarakanov V. E., “On some classes of random variables on cycles of permutations”, Math. USSR-Sb., 36:1 (1980), 87-99.10.1070/SM1980v036n01ABEH001771Search in Google Scholar
[3] Goncharov V. L., “On the field of combinatory analysis”, Amer. Math. Soc. Transl., 19 (1962), 1-46.Search in Google Scholar
[4] Kubilius J., Probabilistic Methods in the Theory of Numbers, Amer. Math. Soc., Providence, R.A., 1964.Search in Google Scholar
[5] Kubilius J., “On the estimate of the second central moment for arbitrary additive arithmetic functions”, Liet. Mat. Rink., 23 (1983), 110-117.10.1007/BF00968593Search in Google Scholar
[6] Klimavičius J., Manstavičius E., “Turán-Kubilius’ inequality on permutations”, Ann. Univ. Sci. Budapest. Sect. Comput., 48 (2018), 45-51.Search in Google Scholar
[7] Manstavičius E., Stepas V., “Variance of additive functions defined on random assemblies”, Lith. Math. J., 57:2 (2017), 222-235.10.1007/s10986-017-9356-1Search in Google Scholar
[8] Pavlov A. I., “On two classes of permutations with number-theoretic conditions on the lengths of the cycles”, Math. Notes, 62:6 (1997), 739-746.10.1007/BF02355462Search in Google Scholar
[9] Sachkov V. N., Introduction to the combinatorial methods of discrete mathematics, MCCME, Moscow, 2004 (in Russian), 424 pp.Search in Google Scholar
[10] Turán P., “Uber Einige Verallgemeinerungen Eines Satzes von Hardy und Ramanujan”, J. London Math. Soc., 11:2 (1936), 125-13310.1112/jlms/s1-11.2.125Search in Google Scholar
[11] Yakymiv A. L., “Asymptotics of the moments of the number of cycles of a random A-permutation”, Math. Notes, 88:5 (2010), 759-766.10.1134/S0001434610110155Search in Google Scholar
[12] Yakymiv A. L., “Asymptotics with remainder term for moments of the total cycle number of random A-permutation”, Discrete Math. Appl, 31:1 (2021), 51-60.10.1515/dma-2021-0005Search in Google Scholar
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Articles in the same Issue
- Frontmatter
- Single diagnostic tests for inversion faults of gates in circuits over arbitrary bases
- Generalized de Bruijn graphs
- A family of asymptotically independent statistics in polynomial scheme containing the Pearson statistic
- Linear recurrent relations, power series distributions, and generalized allocation scheme
- Variance of the number of cycles of random A-permutation
- Multi-dimensional Kronecker sequences with a small number of gap lengths
Articles in the same Issue
- Frontmatter
- Single diagnostic tests for inversion faults of gates in circuits over arbitrary bases
- Generalized de Bruijn graphs
- A family of asymptotically independent statistics in polynomial scheme containing the Pearson statistic
- Linear recurrent relations, power series distributions, and generalized allocation scheme
- Variance of the number of cycles of random A-permutation
- Multi-dimensional Kronecker sequences with a small number of gap lengths
