A lattice on Dyck paths close to the Tamari lattice

TL;DR

This paper introduces a new lattice structure on Dyck paths related to the Tamari lattice, providing enumeration formulas, structural insights, and a new involution that swaps certain path statistics.

Contribution

It defines a novel poset on Dyck paths, proves it forms a lattice, and offers enumerative and structural results including generating functions and involutions.

Findings

The new poset on Dyck paths forms a lattice.

Derived generating functions for counting Dyck paths and intervals.

Identified a new involution swapping path statistics.

Abstract

We introduce a new poset structure on Dyck paths where the covering relation is a particular case of the relation inducing the Tamari lattice. We prove that the transitive closure of this relation endows Dyck paths with a lattice structure. We provide a trivariate generating function counting the number of Dyck paths with respect to the semilength, the numbers of outgoing and incoming edges in the Hasse diagram. We deduce the numbers of coverings, meet and join irreducible elements. As a byproduct, we present a new involution on Dyck paths that transports the bistatistic of the numbers of outgoing and incoming edges into its reverse. Finally, we give a generating function for the number of intervals, and we compare this number with the number of intervals in the Tamari lattice.