Lattice paths inside a table, I

Abstract

A lattice path in ${\mathbb{Z}}^d$ is a sequence ${\nu }_1,{\nu }_2,\dots ,{\nu }_k\in {\mathbb{Z}}^d$ such that the steps ${\nu }_i-{\nu }_{i-1}$ lie in a subset $\mathbf{S}$ of ${\mathbb{Z}}^d$ for all $i=2,\dots ,k$. 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.

Cited by in corpus (1)

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