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Teor. Veroyatnost. i Primenen., 1972, Volume 17, Issue 4, Pages 679–694 (Mi tvp4341)  

This article is cited in 18 scientific papers (total in 18 papers)

Mappings of a finite set with limitations on contours and height

V. N. Sachkov


Abstract: Mappings $\sigma\in\mathfrak{S}^h_n(A)$ of a finite set $\mathfrak{A}$ of $n$ elements into itself are considered under the conditions that the orders of the contours of corresponding graphs $\Gamma (\mathfrak{A},\sigma)$ are elements of a set $A$ and the trees $\Gamma(\mathfrak{A},\sigma)$ have the height not exceeding $h$. The generating functions of different characteristics of such mappings as well as the exact and asymptotic number of such mappings as $n\to\infty$ are found. For the uniform distributions on $\mathfrak{S}^h_n(A)$ with $A$ finite and $n\to\infty$ the distributions of the number of cyclic elements and components in a random mapping are proved to be asymptotically normal. It is shown that the number of free trees in a random forest with the numbers of vertices in trees which are elements of a finite sequence $A$ and the number of cycles in a random solution of the equation $X^d=E$ in the symmetrical group $S_n$ are also asymptotically normal.

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English version:
Theory of Probability and its Applications, 1973, 17:4, 640–656

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Received: 24.12.1970

Citation: V. N. Sachkov, “Mappings of a finite set with limitations on contours and height”, Teor. Veroyatnost. i Primenen., 17:4 (1972), 679–694; Theory Probab. Appl., 17:4 (1973), 640–656

Citation in format AMSBIB
\by V.~N.~Sachkov
\paper Mappings of a finite set with limitations on contours and height
\jour Teor. Veroyatnost. i Primenen.
\yr 1972
\vol 17
\issue 4
\pages 679--694
\jour Theory Probab. Appl.
\yr 1973
\vol 17
\issue 4
\pages 640--656

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    This publication is cited in the following articles:
    1. Yu. V. Bolotnikov, V. N. Sachkov, V. E. Tarakanov, “Asymptotic normality of some variables connected with the cyclic structure of random permutations”, Math. USSR-Sb., 28:1 (1976), 107–117  mathnet  crossref  mathscinet  zmath  isi
    2. A. L. Yakymiv, “On the distribution of the $m$th maximal cycle lengths of random $A$-permutations”, Discrete Math. Appl., 15:5 (2005), 527–546  mathnet  crossref  crossref  mathscinet  zmath  elib
    3. A. L. Yakymiv, “Random $A$-Permutations: Convergence to a Poisson Process”, Math. Notes, 81:6 (2007), 840–846  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    4. A. L. Yakymiv, “Limit theorem for the general number of cycles in a random $A$-permutation”, Theory Probab. Appl., 52:1 (2008), 133–146  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    5. A. L. Yakymiv, “Limit Theorem for the Middle Members of Ordered Cycle Lengths in Random $A$-Permutations”, Theory Probab. Appl., 54:1 (2010), 114–128  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    6. I. Yu. Osmolovskii, “On Estimating Approximation Exactness for Asymptotic Expansions in Polynomial Cases”, Theory Probab. Appl., 54:1 (2010), 154–160  mathnet  crossref  crossref  mathscinet  zmath  isi
    7. A. L. Yakymiv, “On the Number of $A$-Mappings”, Math. Notes, 86:1 (2009), 132–139  mathnet  crossref  crossref  mathscinet  zmath  isi
    8. A. L. Yakymiv, “A limit theorem for the logarithm of the order of a random $A$-permutation”, Discrete Math. Appl., 20:3 (2010), 247–275  mathnet  crossref  crossref  mathscinet  zmath  elib  elib
    9. A. L. Yakymiv, “Asymptotics of the Moments of the Number of Cycles of a Random $A$-Permutation”, Math. Notes, 88:5 (2010), 759–766  mathnet  crossref  crossref  mathscinet  isi
    10. V. N. Sachkov, “Sluchainye otobrazheniya s nepodvizhnymi elementami”, Matem. vopr. kriptogr., 2:2 (2011), 95–118  mathnet  crossref
    11. A. L. Yakymiv, “Random $A$-permutations and Brownian motion”, Proc. Steklov Inst. Math., 282 (2013), 298–318  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    12. A. L. Yakymiv, “On the number of cyclic points of random $A$-mapping”, Discrete Math. Appl., 23:5-6 (2013), 503–515  mathnet  crossref  crossref  mathscinet  elib  elib
    13. A. L. Yakymiv, “On a number of components in a random $A$-mapping”, Theory Probab. Appl., 59:1 (2015), 114–127  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    14. A. L. Yakymiv, “On the Number of Components of Fixed Size in a Random $A$-Mapping”, Math. Notes, 97:3 (2015), 468–475  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    15. A. L. Yakymiv, “Limit theorems for the logarithm of the order of a random $A$-mapping”, Discrete Math. Appl., 27:5 (2017), 325–338  mathnet  crossref  crossref  mathscinet  isi  elib
    16. A. L. Yakymiv, “On the order of random permutation with cycle weights”, Theory Probab. Appl., 63:2 (2018), 209–226  mathnet  crossref  crossref  mathscinet  isi  elib
    17. A. L. Yakymiv, “Asimptotika momentov chisla tsiklov sluchainoi $A$-podstanovki s ostatochnym chlenom”, Diskret. matem., 31:3 (2019), 114–127  mathnet  crossref  mathscinet
    18. A. L. Yakymiv, “Raspredelenie ob'ema naibolshei komponenty sluchainogo $A$-otobrazheniya”, Diskret. matem., 31:4 (2019), 116–127  mathnet  crossref
  • Теория вероятностей и ее применения Theory of Probability and its Applications
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