A Bijection on Bilateral Dyck Paths

Abstract

It is known that both the number of Dyck paths with 2n steps and k peaks, and the number of Dyck paths with 2n steps and k steps at odd height follow the Narayana distribution. In this paper we present a bijection which explicitly illustrates this equinumeracy. Moreover, we extend this bijection to bilateral Dyck paths. The restriction to Dyck paths preserves the number of contacts.