Counting Humps in Motzkin paths

Details Counting Humps in Motzkin paths Counting Humps in Motzkin paths

2011-09-13

arxiv.org/abs/1109.2661v1

Mathematics

Combinatorics

8 pages, 2 Figures

Scientific paper

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.

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