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The paper introduces the concept of generalized Catalan numbers as a useful tool for solving many theoretical and applied combinatorial-probabilistic problems. Combined with analytical transformation software algorithms, the generalized Catalan numbers simplify the solution of many problems in computer science and applied mathematics. In particular, they turn out to be an effective tool for solving problems related to the registration of random-dot images, in signal transformations of various degrees of smoothness, and when developing speed-optimal algorithms for searching for pulsed-point objects with a random generation time of super short pulses. The proposal of the authors to formulate the problems of enumerative combinatorics in a word-symbolic form naturally leads to multidimensional extensions of the classical Catalan numbers and has several advantages. The combined use of multidimensional Catalan numbers and high-performance computer algebra systems enables solving a number of complex applied problems related to the reliability of the registration of random-dot images.

The paper discusses an approach to solving complex probabilistic combinatorial problems. The approach is based on the use of specialized software systems for analytical transformations for computing systems. One of the problems associated with the partition of the interval (which arises in the study of the reliability of reading discrete-point fields and digital images) is considered in the paper, and new previously unknown analytical formulas have been successfully obtained. The efficiency of the developed software systems is ensured by two factors: firstly, the development of high-speed specialized recursive-combinatorial algorithms; secondly, the software implementation on high-performance computing clusters using modern programming and development tools (such as C ++ and MPI).
Examples of particular solutions to the described problem that are obtained with the help of constructed computer systems are presented. An effective approach to solving complex probabilistic combinatorial problems is demonstrated. For the proposed approach, a computer is not just a powerful "calculator", but is an effective assistant with a wide range of algorithms and programs for complex and branched analytical transformations.