%0 Journal Article
%T Counting Humps in Motzkin paths
%A Yun Ding
%A Rosena R. X. Du
%J Mathematics
%D 2011
%I arXiv
%X In this paper we study the number of humps (peaks) in Dyck, Motzkin and Schr\"{o}der 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 Schr\"{o}der paths, we also get an identity involving products of binomial coefficients.
%U http://arxiv.org/abs/1109.2661v1