Lattice paths inside a table, I

TL;DR

This paper investigates lattice paths within a grid with specific step sets, deriving explicit formulas for certain cases, proving a conjecture related to these counts, and exploring connections to Fibonacci and Pell-Lucas numbers.

Contribution

It provides explicit formulas for lattice path counts in a grid with particular steps, proves a conjecture by Povolotsky, and links these counts to well-known number sequences.

Findings

Explicit formulas for $ ext{I}_m(n)$ in special cases

Proof of Povolotsky's conjecture for $ ext{I}_n(n)$

Relationships established between lattice paths and Fibonacci, Pell-Lucas numbers

Abstract

A lattice path in $\mathbb{Z}^d$ is a sequence $\nu_1,\nu_2,\ldots,\nu_k\in\mathbb{Z}^d$ such that the steps $\nu_i-\nu_{i-1}$ lie in a subset $\mathbf{S}$ of $\mathbb{Z}^d$ for all $i=2,\ldots,k$. Let $T_{m,n}$ be the $m\times n$ table in the first area of the $xy$-axis and put $\mathbf{S}=\{(1,1),(1,0),(1,-1)\}$. Accordingly, let $\mathcal{I}_m(n)$ denote the number of lattice paths starting from the first column and ending at the last column of $T$. We will study the numbers $\mathcal{I}_m(n)$ and give explicit formulas for special values of $m$ and $n$. As a result, we prove a conjecture of \textit{Alexander R. Povolotsky} involving $\mathcal{I}_n(n)$. Finally, we present some relationships between the number of lattice paths and Fibonacci and Pell-Lucas numbers, and pose an open problem.