A Counting Function

TL;DR

This paper introduces a new counting function related to binomial coefficients, providing explicit formulas, recurrence relations, and demonstrating its application across various combinatorial and mathematical structures.

Contribution

The paper defines a novel counting function, derives explicit formulas, recurrence relations, and links it to numerous mathematical and combinatorial objects, expanding understanding of these counts.

Findings

Derived explicit formulas for the counting function.

Established recurrence relations satisfied by the function.

Connected the function to various combinatorial and number theoretic structures.

Abstract

We define a counting function that is related to the binomial coefficients. An explicit formula for this function is proved. In some particular cases, simpler explicit formuls are derived. We also derive a formula for the number of (0,1)-matrices, having a fixed number of 1's, and having no zero rows and zero columns. Further, we show that our function satisfies several recurrence relations. The relationship of our counting function with different classes of integers is then examined. These classes include: different kind of figurate numbers, the number of points on the surface of a square pyramid, the magic constants, the truncated square numbers, the coefficients of the Chebyshev polynomials, the Catalan numbers, the Dellanoy numbers, the Sulanke numbers, the numbers of the coordination sequences, and the number of the crystal ball sequences of a cubic lattice. In the last part of the paper, we prove that several configurations are counted by our function. Some of these are: the number of spanning subgraphs of the complete bipartite graph, the number of square containing in a square, the number of coloring's of points on a line, the number of divisors of some particular numbers, the number of all parts in the compositions of an integer, the numbers of the weak compositions of integers, and the number of particular lattice paths. We conclude by counting the number of possible moves of the rook, bishop, and queen on a chessboard. The most statements in the paper are provided by bijective proofs in terms of insets, which are defined in the paper. With this we want to show that different configurations may be counted by the same method.