A bijection for Delannoy paths

TL;DR

This paper presents a bijection between two classes of lattice paths, connecting central Delannoy paths to a broader class with slope restrictions, revealing structural similarities.

Contribution

It introduces a novel bijection linking central Delannoy paths to slope-restricted paths, enhancing combinatorial understanding.

Findings

Established a bijection between Delannoy paths and slope-restricted paths.

Revealed structural correspondence between different lattice path classes.

Potential applications in combinatorics and path enumeration.

Abstract

We exhibit a bijection between central Delannoy $n$-paths, that is, lattice paths from the origin to $(n,n)$ with steps $E=(1,0), \,N=(0,1),\,D=(1,1)$ and the lattice paths from the origin to $(n+1,n)$ where the only restriction on the steps is that they have finite nonnegative slope.