Bijective counting of humps and peaks in $(k,a)$-paths

TL;DR

This paper provides bijective proofs linking the total number of humps and peaks in all $(k,a)$-paths of order $n$ to the count of super $(k,a)$-paths, extending previous combinatorial results.

Contribution

It offers the first bijective proofs for the relations between humps, peaks, and super $(k,a)$-paths, generalizing earlier findings.

Findings

Established bijective proofs for hump and peak relations.

Extended previous results to general $(k,a)$-paths.

Connected hump and peak counts to super-paths in a bijective manner.

Abstract

Recently, Mansour and Shattuck related the total number of humps in all of the $(k, a)$-paths of order $n$ to the number of super $(k, a)$-paths, which generalized previous results concerning the cases when $k = 1$ and $a = 1$ or $a = \infty$. They also derived a relation on the total number of peaks in all of the $(k, a)$-paths of order $n$ and the number of super $(k, a)$-paths, and asked for bijective proofs. In this paper, we will give bijective proofs of these two relations.