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

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