On the Number of Walks in a Triangular Domain

TL;DR

This paper derives an explicit generating function for walks within a triangular lattice domain, revealing connections to two-colored Motzkin paths and ballot paths, and explores weighted sublattice structures.

Contribution

It provides a new explicit formula for the generating function of walks in a triangular domain with weighted sublattices, linking combinatorial path models.

Findings

Explicit formula for the generating function of walks in a triangular domain.

Walks starting in a corner are equinumerous with two-colored Motzkin paths.

Connections established between lattice walks and classical combinatorial paths.

Abstract

We consider walks on a triangular domain that is a subset of the triangular lattice. We then specialise this by dividing the lattice into two directed sublattices with different weights. Our central result is an explicit formula for the generating function of walks starting at a fixed point in this domain and ending anywhere within the domain. Intriguingly, the specialisation of this formula to walks starting in a fixed corner of the triangle shows that these are equinumerous to two-coloured Motzkin paths, and two-coloured three-candidate Ballot paths, in a strip of finite height.