Cornerless, peakless, valleyless Motzkin paths (regular and skew) and applications to bargraphs

TL;DR

This paper studies special types of Motzkin paths, including cornerless, peakless, and valleyless paths, extending their enumeration to skew versions and applications to bargraphs using generating functions and the kernel method.

Contribution

It introduces new classifications of Motzkin paths, extends the analysis to skew paths, and provides explicit enumeration formulas using advanced combinatorial techniques.

Findings

Derived explicit generating functions for cornerless, peakless, and valleyless Motzkin paths.

Extended the concept to skew Motzkin paths and counted specific step occurrences.

Established bijections between cornerless Motzkin paths and bargraph prefixes.

Abstract

Motzkin paths consist of up-steps, down-steps, horizontal steps, never go below the $x$-axis and return to the $x$-axis. Versions where the return to the $x$-axis isn't required are also considered. A path is peakless (valleyless) if $UD$ (if $DU$) never occurs. If it is both peakless and valleyless, it is called cornerless. Deutsch and Elizalde have linked cornerless Motzkin paths and bargraphs bijectly. Thus, instead of prefixes of bargraphs one might consider prefixes of cornerless Motzkin paths. In this paper, this is extended by counting the occurrences of $UD$ resp., $DU$. The concepts are extended to so-called skew Motzkin paths. Methods are generating functions and the kernel method to compute explicit forms.