Between Broadway and the Hudson: A Bijection of Corridor Paths

TL;DR

This paper introduces a bijective proof for a generalized combinatorial equivalence between constrained corridor paths, expanding understanding of path enumeration within bounded regions.

Contribution

It provides a new encoding method for lattice paths and establishes a bijection that generalizes known path equinumeracies within a band.

Findings

Establishes a bijection between two classes of constrained lattice paths.

Generalizes the equinumeracy of Dyck paths and their prefixes.

Introduces a new encoding technique for lattice paths.

Abstract

We present a substantial generalization of the equinumeracy of grand Dyck paths and Dyck-path prefixes, constrained within a band. The number of constrained paths starting at level $i$ and ending in a window of size $2j+2$ is equal to the number starting at level $j$ and ending in a window of size $2i+2$ centered around the same point. A new encoding of lattice paths provides a bijective proof.