Counting of paths and the multiplicity of determinantal rings

TL;DR

This paper develops formulas for counting non-intersecting paths in ladder-shaped regions and applies them to provide a new combinatorial proof of Fibonacci numbers, linking path counting to classical number sequences.

Contribution

It introduces new formulas for counting non-intersecting paths in ladder regions and offers a novel combinatorial proof of Fibonacci numbers.

Findings

Derived formulas for non-intersecting path counts in ladder regions

Provided a new combinatorial proof of Fibonacci numbers

Linked path counting with classical number sequences

Abstract

In this paper, we derive several formulas of counting families of non-intersecting paths for two-sided ladder-shaped regions. As an application, we give a new proof to a combinatorial interpretation of Fibonacci numbers obtained by G. Andrews in 1974.