Composition of numbers with constraints and the hierarchical structure of planar sections of Pascal's pyramid

In this paper, we examine compositions of natural numbers with constraints on natural parts and their relationship with hierarchical combinatorial objects. We derive a formula for calculating the number of such compositions with three constraints based on the sums of elements of planar sections of Pascal's pyramid. Also, we obtain recurrence relations and generating functions for the numbers of compositions and examine some important special cases for well-known combinatorial numbers.