Matrices about the lattice paths

TL;DR

This paper explores the use of matrix methods and recursion to analyze four combinatorial numbers related to integer lattice paths, providing explicit formulas and matrix decompositions that differ from previous approaches.

Contribution

It introduces a new matrix-based recursive approach to study combinatorial numbers associated with lattice paths, including explicit formulas and matrix decompositions.

Findings

Derived explicit formulas for the combinatorial numbers.

Presented new matrix decompositions and relations.

Compared with Andre1's method, offering a different perspective.

Abstract

From the matrix point of view, we use the recursion to discuss four combinatorial numbers in terms of the integer lattice paths, this is different from Andr\'a's method (Andra). We give four tables and matrices, and their relations, and give their matrix decompositions. By their relations, we give the explicit formulas of the special combinatorial numbers.