A Note on the Gessel Numbers

TL;DR

This paper explores properties of Gessel numbers, deriving a recurrence relation, providing a new proof for their closed-form formula, and offering a novel combinatorial interpretation.

Contribution

It introduces a new recurrence relation, offers a fresh proof of the Gessel numbers' closed formula, and presents a new combinatorial interpretation.

Findings

Derived a recurrence relation between P(n,r) and P(n-1,r+1)

Provided a new proof for the closed formula of P(n,r)

Presented a new combinatorial interpretation of Gessel numbers

Abstract

The Gessel number $P(n,r)$ represents the number of lattice paths in a plane with unit horizontal and vertical steps from $(0,0)$ to $(n+r,n+r-1)$ that never touch any of the points from the set $\{(x,x)\in \mathbb{Z}^2: x \geq r\}$. In this paper, we use combinatorial arguments to derive a recurrence relation between $P(n,r)$ and $P(n-1,r+1)$. Also, we give a new proof for a well-known closed formula for $P(n,r)$. Moreover, a new combinatorial interpretation for the Gessel numbers is presented.