DESCRIPTION OF A COMBINATORIAL ALGORITHM FOR ENUMERATING ORDERED MULTIPLICATIVE FACTORIZATIONS
The article is devoted to solving a typical combinatorial problem – a systematic enumeration of all objects of a certain type. The decomposition of an integer into an ordered product of integer multipliers (the so-called
-profile of number
) is accepted as an object for enumeration. The purpose of the article is to create a combinatorial algorithm for the systematic enumeration of all
-profiles of number
. To achieve this goal, a formal formulation of the problem was carried out, its analysis was carried out, in which the concept of a characteristic matrix of
-profile
was introduced and a one-to-one correspondence between profiles and characteristic matrices was established, which made it possible to move from enumerating profiles to enumerating characteristic matrices, followed by their transformation into profiles. An
-equivalence relation is introduced on the set of characteristic matrices, which makes it possible to divide it into classes and list the elements of the selected
-equivalence class independently of the rest of the characteristic matrices. It is shown that the generation of an
-equivalent characteristic matrix is the process of determining its elements by distributing units of multiplicities of prime factors in the canonical expansion of a number
, and this process is considered in detail. At the synthesis stage, the results of the task analysis are translated into the desired algorithm. An example of enumeration
-profiles of number
using this algorithm is given.
Rumbesht V.V., Burdanova E.V. Description of a Combinatorial Algorithm for Enumerating Ordered Multiplicative Factorizations // Research result. Information technologies. – T.11, №3, 2026. – P. 100-111. DOI: 10.18413/2518-1092-2026-11-3-0-9
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1. Vinogradov I.M. Fundamentals of Number Theory. Moscow, GITTL, 1952. – 180 p.
2. Giyasi A., Mikhailov I.P., Chubarikov V.N. On the uniform distribution of remainders in the expansion of real numbers according to a multiplicative number system. // Chebyshevskii Sbornik. 2022. No. 23(5). pp. 38-44.
3. Danilov A.V., Makarychev P.P. Combinatorial algorithm for target assignment // Izvestiya Vysshikh Uchebnykh Zavedenii. Povolzhsky Region. Technical Sciences. 2025. No. 3 (75). pp. 86-99.
4. Ivanov B.N. Discrete Mathematics: Algorithms and Programs. 2001. Moscow: Laboratoriya Bazovykh Znaniy. 288 p.
5. Inyutin S.A. Computational Tools for Modular Algorithms: Monograph. Moscow, 2024.
6. Study of approaches to detecting moving objects in noisy images / Abramov K.V., Alexandrov K.S., Balabanova T.N., Babenko A.A., Burdanova E.V. // Research result. Information technologies. – Т. 10, №1, 2025. – P. 47-57.
7. Kaygorodov E.V., Krylov P.A., Tuganbaev A.A. On some linear maps of incidence coalgebras // Applied Mathematics & Physics. – 2024. – T. 56. No. 4. – P. 273-285.
8. Method of System-Object Modeling of Document Flow / Najajra M.H., Bobyshev P.P., Fedorov V.I., Lozovaya S.Y., Babenko A.A. // Economics. Information Technologies. – 2026. – 53(1) – P. 122-135.
9. Chernomorets D.A., Bolgova E.V., Chernomorets A.A., Petina M.A. On estimating the size of informative fragments in the sea surface images // Research result. Information technologies. – Т.9, №2, 2024. – P. 3-11.
10. Polyuga V.A., Shablya Yu.V. Investigation of the Performance of Software Implementations of Combinatorial Generation Algorithms Based on Data Representation Approaches // Applied Mathematics and Informatics: Modern Research in Natural and Technical Sciences. Proceedings of the IX International Scientific and Practical Conference (Workshop-Seminar) of Young Scientists. Tolyatti. – 2023. – P. 60-66.
11. Rumbesht V.V., Burdanova E.V. Combinatorics of ordered multiplicative factorizations. // Scientific Bulletin of Belgorod State University: Economics. Informatics. 2020. – No. 47 (1). – P. 126-134.
12. Ryzhenko K.V., Khachay M.Yu., Neznakhina E.D. Approximation algorithms with constant accuracy bounds for certain asymmetric combinatorial routing problems // Modern Problems of Mathematics and Its Applications. Abstracts of the International (54th All-Russian) Youth School-Conference. Yekaterinburg. – 2023. – P. 56.
13. Sohrabi M., Fathollahi-Fard A.M., Gromov V.A. Genetic Engineering Algorithm (GEA): An Efficient Metaheuristic Algorithm for Solving Combinatorial Optimization Problems // Automation and Remote Control. – 2024. – No. 3. – P. 23-37.
14. Tokareva A.V., Kruchinin D.V. On the Applicability of Combinatorial Generation Algorithms to the Inventory Process // Reshetnev Readings: Proceedings of the XXVII International Scientific and Practical Conference Dedicated to the Memory of Academician M.F. Reshetnev, General Designer of Rocket and Space Systems: in 2 parts. Krasnoyarsk. – 2023. – P. 171-173.
15. Adams S. Locally free actions on lorentz manifolds // Geometric And Functional Analysis. – 2000. – Т. 10. № 3. – P. 453-515.
16. Antinucci G., Giuliani A., Greenblatt R.L. Non-integrable ising models in cylindrical geometry: grassmann representation and infinite volume limit // Annales Henri Poincare. – 2022. – Т. 23. № 3. – P. 1061-1139.
17. Holroyd A.E., Janson S., Wästlund J. Minimal matchings of point processes // Probability Theory and Related Fields. – 2022. – Т. 184. № 1. – P. 571-611.
18. Isaev A.P. Quantum groups and yang-baxter equations // Natural Science Review. – 2025. – Т. 2. № 2. – P. 1-192.
19. Marò S., Bonanno C. Asymptotic behaviour of orbit determination for hyperbolic maps // Celestial Mechanics and Dynamical Astronomy. – 2021. Т. 133. – № 6.
20. Jiang H., Benzaria S., Alsadun N., Jia J., Czaban-Jóźwiak Ju., Guillerm V., Shkurenko A., Thiam Z., Bonneau M., Maka V.K., Chen Zh., Ameuhhr Z.O., O’Keeffe M., Eddaoudi M. Merged-nets enumeration for the systematic design of multicomponent reticular structures // Science. – 2024. – Т. 386. – № 6722. – P. 659-666.