Modeling of a segment’s complete coverings based on the sums of Pascal’s pyramid plane sections elements

Combinatorial objects play an important role in data processing problems, storage, analysis, creation of algorithms and optimization. This work is devoted to the modeling of combinatorial numbers and to the study of some of their geometric properties and interpretations based on a spatial combinatorial configuration called Pascal’s pyramid. Complete coverings of a segment and their relationship with combinatorial objects of a hierarchical structure are considered. A formula for calculating the number of a segment’s complete coverings based on the sums of Pascal’s pyramid plane sections elements is obtained. Recurrence relations and generating functions of the segment’s complete coverings number are found. Various formulas for calculating the number of such coverings are obtained, because of the symmetry of Pascal’s pyramid and some of the most important special cases are considered using the known combinatorial numbers as example.