Kreweras's Narayana number identity has a simple Dyck path interpretation

TL;DR

This paper provides a simple Dyck path interpretation of Kreweras's Narayana number identity, linking peaks and peak plateaus in Dyck paths to combinatorial counts.

Contribution

It introduces a novel combinatorial interpretation of Kreweras's identity, connecting peaks and peak plateaus in Dyck paths.

Findings

Counts Dyck paths with a given number of peaks by peak plateaus

Establishes a direct combinatorial interpretation of Kreweras's identity

Enhances understanding of Narayana numbers in Dyck path combinatorics

Abstract

We show that an identity of Kreweras for the Narayana numbers counts Dyck paths with a given number of peaks by number of peak plateaus, where a peak plateau is a run of consecutive peaks that is immediately preceded by an upstep and followed by a downstep.