Some properties of a new partial order on Dyck paths

TL;DR

This paper introduces a new partial order on Dyck paths, proves it forms meet-semilattices, and uncovers connections with bicubic planar maps and Hochschild polytopes, revealing new combinatorial and geometric insights.

Contribution

It presents a novel partial order on Dyck paths, establishes its algebraic structure, and links it to known combinatorial objects and polytopes, expanding understanding of Dyck path posets.

Findings

Posets are meet-semilattices

Number of intervals matches bicubic planar maps

Connection with Hochschild polytopes

Abstract

We introduce and study a new partial order on Dyck paths. We prove that these posets are meet-semilattices. We show that their numbers of intervals are the same as the number of bicubic planar maps. We describe an unexpected connection with the Hochschild polytopes of Saneblidze.