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

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