A lattice on Dyck paths close to the Tamari lattice

2023-09-01

arXiv.org

AI Summary

This paper introduces a novel partial order on the set of Dyck paths, whose covering relation is a proper subset of that of the Tamari lattice; it proves that the transitive closure forms a lattice—the Dyck lattice—and systematically compares it with the Tamari lattice. Method: Integrating poset theory, bijective combinatorics, and generating function techniques, the authors first define an approximating covering relation for this lattice; construct the first involution exchanging indegree and outdegree in its Hasse diagram; and derive the trivariate generating function enumerating Dyck paths by semilength, outdegree, and indegree. Contribution/Results: The paper provides exact enumeration formulas for covering relations and for join- and meet-irreducible elements; obtains the generating function for the number of intervals; and establishes that the Dyck lattice has strictly more intervals than the Tamari lattice—quantifying a fundamental structural complexity gap between the two lattices.

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.

Problem

• Introduces new poset structure on Dyck paths
• Analyzes generating functions for path statistics
• Compares intervals with Tamari lattice counts

Innovation

• New poset structure on Dyck paths
• Trivariate generating function for counting
• New involution on Dyck paths