Partial Skew Motzkin Paths

TL;DR

This paper extends the enumeration of skew Motzkin paths by allowing paths to end at a prescribed level, using generating functions and the kernel method, and provides asymptotic analysis of their total count and average height.

Contribution

It introduces a new enumeration framework for skew Motzkin paths ending at specific levels using the kernel method and generating functions.

Findings

Extended enumeration to paths ending at prescribed levels

Derived generating functions for skew Motzkin paths

Provided asymptotic formulas for total count and average height

Abstract

Motzkin paths consist of up-steps, down-steps, level-steps, and never go below the $x$-axis. They return to the $x$-axis at the end. The concept of skew Dyck path \cite{Deutsch-italy} is transferred to skew Motzkin paths, namely, a left step $(-1,-1)$ is additionally allowed, but the path is not allowed to intersect itself. The enumeration of these combinatorial objects was known \cite{Qing}; here, using the kernel method, we extend the results by allowing them to end at a prescribed level $j$. The approach is completely based on generating functions. Asymptotics of the total number of objects as well as the average height are also given.