  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
$\frac{1}{k}{n-1\choose k-1}{n\choose k-1}$.
