# Geometric realizations of Tamari interval lattices via cubic coordinates

## TL;DR

This paper introduces cubic coordinates as a new combinatorial tool to represent Tamari interval lattices, providing geometric realizations and analyzing their structural properties.

## Contribution

It defines cubic coordinates, establishes their bijection with Tamari intervals, and demonstrates their geometric and lattice properties, including shellability.

## Key findings

- Cubic coordinates form a lattice isomorphic to Tamari intervals.
- Geometric realizations of these lattices are constructed and analyzed.
- The poset of cubic coordinates is shellable.

## Abstract

We introduce cubic coordinates, which are integer words encoding intervals in the Tamari lattices. Cubic coordinates are in bijection with interval-posets, themselves known to be in bijection with Tamari intervals. We show that in each degree the set of cubic coordinates forms a lattice, isomorphic to the lattice of Tamari intervals. Geometric realizations are naturally obtained by placing cubic coordinates in space, highlighting some of their properties. We consider the cellular structure of these realizations. Finally, we show that the poset of cubic coordinates is shellable.