Product-Coproduct Prographs and Triangulations of the Sphere

TL;DR

This paper introduces product-coproduct prographs as a new combinatorial framework that generalizes classical Catalan structures to three dimensions, specifically explaining sphere triangulations and extending the Tamari lattice.

Contribution

It presents a novel three-dimensional generalization of Catalan objects using product-coproduct prographs and extends the Tamari lattice to these structures.

Findings

Product-coproduct prographs effectively model sphere triangulations.

A natural extension of the Tamari lattice to three-dimensional structures.

Unified combinatorial framework for classical and higher-dimensional Catalan objects.

Abstract

In this paper, we explain how the classical Catalan families of objects involving paths, tableaux, triangulations, parentheses configurations and more generalize canonically to a three-dimensional version. In particular, we present product-coproduct prographs as central objects explaining the combinatorics of the triangulations of the sphere. Then we expose a natural way to extend the Tamari lattice to the product-coproduct prographs.