Enumerations of humps and peaks in $(k,a)$-paths and $(n,m)$-Dyck paths via bijective proofs

TL;DR

This paper provides bijective proofs for formulas relating peaks in $(k,a)$-paths and extends these results to $(n,m)$-Dyck paths, offering new combinatorial insights and generalizations.

Contribution

It offers the first bijective proofs of Mansour and Shattuck's formulas and extends the analysis to coprime $(n,m)$-Dyck paths, generalizing known peak enumeration results.

Findings

Bijective proofs of formulas for peaks in $(k,a)$-paths.

A bijection relating peaks in $(n,m)$-Dyck paths to free paths.

Enumeration of $(n,m)$-Dyck paths with a fixed number of peaks.

Abstract

Recently Mansour and Shattuck studied $(k,a)$-paths and gave formulas that relate the total number of humps (peaks) in all $(k,a)$-paths to the number of super $(k,a)$-paths. These results generalize earlier results of Regev on Dyck paths and Motzkin paths. Their proofs are based on generating functions and they asked for bijective proofs for their results. In this paper we first give bijective proofs of Mansour and Shattuck's results, then we extend our study to $(n,m)$-Dyck paths. We give a bijection that relates the total number of peaks in all $(n,m)$-Dyck paths to certain free $(n,m)$-paths when $n$ and $m$ are coprime. From this bijection we get the number of $(n,m)$-Dyck paths with exactly $j$ peaks, which is a generalization of the well-known result that the number Dyck paths of order $n$ with exactly $j$ peaks is the Narayana number $\frac{1}{k}{n-1\choose k-1}{n\choose k-1}$.