A Bijection on Bilateral Dyck Paths

TL;DR

This paper introduces a bijection that demonstrates the equinumeracy of Dyck paths with specific peak and height properties, extending it to bilateral Dyck paths while preserving contacts.

Contribution

It presents a new bijection illustrating the equinumeracy and extends it to bilateral Dyck paths, preserving contacts.

Findings

Bijection explicitly illustrates equinumeracy.

Extension to bilateral Dyck paths preserves contacts.

Confirms Narayana distribution for these 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.