Modeling of a segment’s incomplete coverings based on the sums of Pascal’s pyramid upper units elements

Combinatorial objects are one of the most important tools for solving problems related to data analysis and storage, creation and optimization of algorithms for their processing. In this case, there is a need to model both individual combinatorial numbers and their sequences. In this paper we study some geometric properties and interpretations of the Pascal’s pyramid upper units, which is a spatial combinatorial configuration of a hierarchical structure. We obtaine the generalized sequence of combinatorial numbers and present a calculation formula in explicit form for each of this number, as well as a recurrence relation and a generating function. We prove that the sum of the Pascal’s pyramid upper unit elements is equal to the number of incomplete covers of the corresponding segment. Due to the symmetry of Pascal’s pyramid we obtaine some various formulas for calculating the number of such covers, and consider some of the most important special cases using the example of known combinatorial numbers.