Counting Humps in Motzkin paths

TL;DR

This paper investigates the number of peaks in Dyck, Motzkin, and Schr"{o}der paths, providing bijective proofs and new combinatorial identities related to these path structures.

Contribution

It offers a bijective proof for the relation between peaks in Motzkin paths and super Dyck paths, and introduces new identities involving these paths.

Findings

Number of peaks in Dyck paths relates to Narayana numbers.

Established bijections between peaks and super paths.

Derived new binomial coefficient identities from path counts.

Abstract

In this paper we study the number of humps (peaks) in Dyck, Motzkin and Schr\"{o}der paths. Recently A. Regev noticed that the number of peaks in all Dyck paths of order $n$ is one half of the number of super Dyck paths of order $n$. He also computed the number of humps in Motzkin paths and found a similar relation, and asked for bijective proofs. We give a bijection and prove these results. Using this bijection we also give a new proof that the number of Dyck paths of order $n$ with $k$ peaks is the Narayana number. By double counting super Schr\"{o}der paths, we also get an identity involving products of binomial coefficients.