# Classification of uniform flag triangulations of the boundary of the full root polytope of type $A$

## TL;DR

This paper classifies all uniform flag triangulations of the boundary of the full root polytope of type A, revealing three natural classes and expressing face counts through Catalan, Delannoy, and Bessel functions.

## Contribution

It provides a complete classification of uniform flag triangulations of the full root polytope of type A and derives their refined face counts using special generating functions.

## Key findings

- Identified three classes of triangulations: lex, revlex, and Simion.
- Derived face count formulas involving Catalan, Delannoy, and Bessel functions.
- Established that refined face counts depend only on the triangulation class.

## Abstract

The full root polytope of type $A$ is the convex hull of all pairwise differences of the standard basis vectors which we represent by forward and backward arrows. We completely classify all flag triangulations of this polytope that are uniform in the sense that the edges may be described as a function of the relative order of the indices of the four basis vectors involved. These fifteen triangulations fall naturally into three classes: three in the lex class, three in the revlex class and nine in the Simion class. We also consider a refined face count where we distinguish between forward and backward arrows. We prove the refined face counts only depend on the class of the triangulations. The refined face generating functions are expressed in terms of the Catalan and Delannoy generating functions and the modified Bessel function of the first kind.