Two kinds of partial Motzkin paths with air pockets

TL;DR

This paper introduces two new classes of partial Motzkin paths with air pockets, providing their enumeration, properties, and connections to Motzkin numbers, using combinatorial and Riordan array techniques.

Contribution

It presents two novel generalizations of Motzkin paths with air pockets, including enumeration formulas and Riordan array representations.

Findings

Derived enumeration formulas for the new path classes

Connected the new paths to Motzkin numbers

Expressed results using Riordan arrays

Abstract

Motzkin paths with air pockets (MAP) are defined as a generalization of Dyck paths with air pockets by adding some horizontal steps with certain conditions. In this paper, we introduce two generalizations. The first one consists of lattice paths in $\Bbb{N}^2$ starting at the origin made of steps $U=(1,1)$, $D_k=(1,-k)$, $k\geq 1$ and $H=(1,0)$, where two down steps cannot be consecutive, while the second one are lattice paths in $\Bbb{N}^2$ starting at the origin, made of steps $U$, $D_k$ and $H$, where each step $D_k$ and $H$ is necessarily followed by an up step, except for the last step of the path. We provide enumerative results for these paths according to the length, the type of the last step, and the height of its end-point. A similar study is made for these paths read from right to left. As a byproduct, we obtain new classes of paths counted by the Motzkin numbers. Finally, we express our results using Riordan arrays.