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Recently Mansour and Shattuck studied $(k,a)$-paths and gave formulas thatrelate the total number of humps (peaks) in all $(k,a)$-paths to the number ofsuper $(k,a)$-paths. These results generalize earlier results of Regev on Dyckpaths and Motzkin paths. Their proofs are based on generating functions andthey asked for bijective proofs for their results. In this paper we first givebijective 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 peaksin all $(n,m)$-Dyck paths to certain free $(n,m)$-paths when $n$ and $m$ arecoprime. From this bijection we get the number of $(n,m)$-Dyck paths withexactly $j$ peaks, which is a generalization of the well-known result that thenumber Dyck paths of order $n$ with exactly $j$ peaks is the Narayana number$\frac{1}{k}{n-1\choose k-1}{n\choose k-1}$.

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