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

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.

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