Volume 13, no. 3Pages 103 - 115

Probabilistic Models of Combinatorial Schemes

N.Yu. Enatskaya
An enumerative method is proposed for the analysis of combinatorial schemes in the pre-asymptotic region of variation of their parameters based on the construction of their probabilistic mathematical model, which represents for each scheme an iterative random process of sequential non-repeated formation of all its outcomes with a certain discipline of their numbering by unitary addition of certain elements of the scheme to a given value in it. Due to the importance for a number of studies of the scheme of recurrence of listing its outcomes, if it does not lie in its nature, it can be achieved by introducing into the scheme some restrictions that do not lead to a change in their set, do not change their probability and should be taken into account. The design of the process under the appropriate conditions of each scheme is graphically depicted by a graph with the probabilities of iterative transitions specified in it, which determine the final distribution on the set of its outcomes. On this basis, the problems of determining the number of outcomes of a scheme, establishing a one-to-one correspondence between numbers and types of its outcomes, called the numbering problem in direct and reverse statements, and finding the probability distribution of all its final outcomes are solved, which makes it possible to model them with the found distribution of playing out the outcome number and the subsequent determination of its modeled form by the result of solving the direct numbering problem. In the absence of an explicit formula for the number of outcomes of a scheme under certain conditions, an estimate of it can be obtained from the results of their modeling, followed by refinement of the numbering problem. The study of models of combinatorial schemes on random processes with the introduction of probabilistic parameters expands the possibilities of their use. The results of the analysis of schemes can be of a nature from numerical methods and algorithms to analytical in the form of recurrence relations and explicit formulas.
Full text
enumeration method; graph method; probabilistic models; numbering problem; rapid modelling.
1. Riordan D. Introduction to Combinatorial Analysis. N.Y., John Wiley, 1958.
2. Sachkov V.N. Vvedenie v kombinatornye metody diskretnoy matematiki [Introduction to Combinatorial Methods of Discrete Mathematics]. Moscow, Moscow Center for Continuous Mathematical Education, 1982. (in Russian)
3. Kolchin V.F., Sevastyanov B.A., Chistyakov V.P. Sluchaynye razmeshcheniya [Random Postings]. Moscow, Nauka, 1976. (in Russian)
4. Knuth D. Iskusstvo programmirovaniya na EVM [The Art of Computer Programming]. Moscow, Mir, 1976. (in Russian)
5. Enatskaya N.Yu. Analysis Combinatorial Schemes in the Pre-Asymptotic Region of Parameter Change. Proceedings of the Kola Science Center of the Russian Academy of Science, 2018, no. 7, p. 117-133. (in Russian)