Bijections between walks inside a triangular domain and Motzkin paths of bounded amplitude

TL;DR

This paper establishes explicit bijections between triangular lattice walks and bounded Motzkin paths, extending to higher dimensions and providing new combinatorial insights with efficient algorithms.

Contribution

It introduces several new bijections connecting triangular walks and Motzkin paths, including extensions to walks starting inside the triangle and higher-dimensional analogs.

Findings

Multiple bijections with linear-time complexity

Extension of results to higher dimensions

Discovery of a bijection between 3D pyramid walks and 2D waffle-shaped walks

Abstract

This paper solves an open question of Mortimer and Prellberg asking for an explicit bijection between two families of walks. The first family is formed by what we name triangular walks, which are two-dimensional walks moving in six directions (0{\deg}, 60{\deg}, 120{\deg}, 180{\deg}, 240{\deg}, 300{\deg}) and confined within a triangle. The other family is comprised of two-colored Motzkin paths with bounded height, in which the horizontal steps may be forbidden at maximal height. We provide several new bijections. The first one is derived from a simple inductive proof, taking advantage of a $2^n$-to-one function from generic triangular walks to triangular walks only using directions 0{\deg}, 120{\deg}, 240{\deg}. The second is based on an extension of Mortimer and Prellberg's results to triangular walks starting not only at a corner of the triangle, but at any point inside it. It has a linear-time complexity and is in fact adjustable: by changing some set of parameters called a scaffolding, we obtain a wide range of different bijections. Finally, we extend our results to higher dimensions. In particular, by adapting the previous proofs, we discover an unexpected bijection between three-dimensional walks in a pyramid and two-dimensional simple walks confined in a bounded domain shaped like a waffle.