Peakless Motzkin paths of bounded height

TL;DR

This paper studies peakless Motzkin paths confined within a bounded height, providing enumeration formulas and asymptotic average height estimates using advanced combinatorial methods.

Contribution

It introduces new enumeration results for peakless Motzkin paths within a strip and derives asymptotic average height formulas.

Findings

Enumeration formulas for peakless Motzkin paths in a strip

Asymptotic average height of such paths

Application of kernel method and singularity analysis

Abstract

There was recent interest in Motzkin paths without peaks (peak: up-step followed immediately by down-step); additional results about this interesting family is worked out. The new results are the enumeration of such paths that live in a strip $[0..\ell]$, and as consequence the asymptotics of the average height, which is given by $2\cdot 5^{-1/4}\sqrt{\pi n}$. Methods include the kernel method and singularity analysis of generating functions.