Counting humps in Motzkin paths

In this paper we study the number of humps (peaks) in Dyck, Motzkin and Schröder paths. Recently A. Regev noticed that the number of peaks in all Dyck paths of order nn is one half of the number of super-Dyck paths of order nn. He also computed the number of humps in Motzkin paths and found a similar relation, and asked for bijective proofs. We give a bijection and prove these results. Using this bijection we also give a new proof that the number of Dyck paths of order nn with kk peaks is the Narayana number. By double counting super-Schröder paths, we also get an identity involving products of binomial coefficients.

► We give a bijection between humps of Motzkin paths and super-Motzkin paths.
► Using this bijection we count the number of humps in all Motzkin paths of order nn.
► We give a new proof that the number of Dyck paths of order nn with kk peaks is the Narayana number.
► We also count the number of humps in Schröder paths, and get a combinatorial proof of an identity.