Lattice paths inside a table, II

TL;DR

This paper derives explicit formulas for counting lattice paths within an m-by-n table, considering paths with specific step constraints and starting/ending points, extending combinatorial path enumeration methods.

Contribution

It provides new explicit formulas for counting lattice paths with particular step patterns and boundary conditions inside a rectangular grid.

Findings

Explicit formulas for ^1(s,t) and (s,t) are derived.

Formulas for the total number of paths from the first to the last column are established.

The results extend combinatorial enumeration techniques for lattice paths.

Abstract

Consider an $m\times n$ table $T$ and latices paths $\nu_1,\ldots,\nu_k$ in $T$ such that each step $\nu_{i+1}-\nu_i=(1,1)$, $(1,0)$ or $(1,-1)$. The number of paths from the $(1,i)$-blank (resp. first column) to the $(s,t)$-blank is denoted by $\mathcal{D}^i(s,t)$ (resp. $\mathcal{D}(s,t)$). Also, the number of all paths form the first column to the las column is denoted by $\mathcal{I}_m(n)$. We give explicit formulas for the numbers $\mathcal{D}^1(s,t)$ and $\mathcal{D}(s,t)$.