Catalan lattices on series parallel interval orders

TL;DR

This paper introduces a unified poset-based framework to describe Dyck and Tamari lattices, simplifying proofs of their relationships and connecting them to Bruhat orders on 312-avoiding permutations.

Contribution

It provides a novel poset-theoretic approach to unify and simplify the understanding of Catalan lattices and their relation to permutation orders.

Findings

Dyck order refines Tamari order.

Dyck and Tamari lattices are isomorphic to certain Bruhat orders.

Simplified proofs of known lattice relationships.

Abstract

Using the notion of series parallel interval order, we propose a unified setting to describe Dyck lattices and Tamari lattices (two well known lattice structures on Catalan objects) in terms of basic notions of the theory of posets. As a consequence of our approach, we find an extremely simple proof of the fact that the Dyck order is a refinement of the Tamari one. Moreover, we provide a description of both the weak and the strong Bruhat order on 312-avoiding permutations, by recovering the proof of the fact that they are isomorphic to the Tamari and the Dyck order, respectively; our proof, which simplifies the existing ones, relies on our results on series parallel interval orders.