2025 , Volume 30, № 2, p.100-111

Purpose. The purpose of the research is to systematize and find an explicit analytical form of generalized Catalan numbers based on a word-symbolic representation, as well as their application for applied problems related to the processing of random point images.

Methodology. An original mechanism for reducing solvable continuous problems to their discrete analogues is proposed. In the practical implementation of the chosen research scheme, there are two types of problems, namely, discrete-combinatorial problems leading to generalized Catalan numbers and problems of calculating multidimensional integrals over convex polyhedra in n-dimensional space requiring the development of high-speed computer algebra programs.

The main content. A generalization of the classical Catalan numbers to the multidimensional case is proposed, based on word-symbolic statistics using a limited alphabet. Probabilistic-combinatorial problems are explicitly formulated; their solution leads to a two-dimensional and then to a three dimensional generalization of the classical Catalan sequence. The main advantages of the new extension of Catalan numbers are the ease of generalization of word-symbolic problems to the multidimensional case and the unity and universality of their formulations.

Findings. Word-symbolic problems leading to multidimensional generalized Catalan numbers are formulated. An explicit form of generalized Catalan numbers in two-dimensional and three dimensional cases is obtained. A number of probabilistic problems related to the analysis of random point images have been solved.

Scientific novelty and originality. The originality and scientific novelty of the article contains both the obtained results and the developed methods to solve them. The article presents new previously unknown probabilistic formulas required for solving problems related to the registration and analysis of random point images. A package of programs has been created for high-speed integrating multidimensional integral expressions over areas limited by a system of hyperplanes in n-dimensional space.