The method of creative telescoping
Journal of Symbolic Computation
Discrete Mathematics - Special issue on selected papers in honor of Henry W. Gould
Enumerative Combinatorics: Volume 1
Enumerative Combinatorics: Volume 1
Note: Counting humps and peaks in generalized Motzkin paths
Discrete Applied Mathematics
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In this paper we study the number of humps (peaks) in Dyck, Motzkin and Schroder paths. Recently A. Regev noticed that the number of peaks in all Dyck paths of order n is one half of the number of super-Dyck paths of order n. 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 n with k peaks is the Narayana number. By double counting super-Schroder paths, we also get an identity involving products of binomial coefficients.