A bijection between two subfamilies of Motzkin paths

TL;DR

This paper presents a direct bijection between two specific subfamilies of Motzkin paths that have the same number of each step type, without relying on auxiliary structures like ternary trees.

Contribution

The authors construct a new, direct bijection between two subfamilies of Motzkin paths, avoiding the use of auxiliary objects such as ternary trees.

Findings

Establishes a direct bijection between two Motzkin path subfamilies.

Shows the two subfamilies are equinumerous without auxiliary structures.

Provides a new combinatorial insight into Motzkin paths.

Abstract

Two subfamilies of Motzkin paths, with the same numbers of up, down, horizontal steps were known to be equinumerous with ternary trees and related objects. We construct a bijection between these two families that does not use any auxiliary objects, like ternary trees.