Application of hierarchical structures based on binary matrices with the generalized arithmetic of Pascal’s triangle in route building problems

Abstract

The paper describes a method for constructing binary matrices based on Pascal’s triangle. It provides the definitions of vertical and horizontal generatrices, consisting of binary elements, and the recurrent rule of Pascal’s triangle, with the help of which a binary matrix is formed. The possibility of parametrization of generatrices is shown by identifying a combinatorial root word, called a generatrix pattern, in terms of word combinatorics. A comparative analysis is provided with the method of generalizing Pascal’s triangle using reduction modulo a prime. Combinatorial differences are shown. Discrete mathematical objects (an integer lattice and combinatorial lattice paths) are described. The paper provides the definitions of lattice paths and describes specific types of paths - the Motzkin paths and the McMahon paths. It gives some combinatorial properties of the Motzkin paths. A method for mapping a binary matrix of the Pascal triangle type to the set of points of an integer combinatorial lattice is described. A method for constructing and predicting the qualitative characteristics of navigation routes by combining binary matrices and integer lattices is developed. Possibilities of parametrization of a binary matrix and an integer lattice, taking into account practical problems, are presented.