Classification Of Uniform Flag Triangulations Of The Boundary Of The Full Root Polytope Of Type A

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 this polytopethat are uniform in the sense that the edges may be described as a function ofthe relative order of the indices of the four basis vectors involved. These fifteentriangulations fall naturally into three classes: three in the lex class, three in therevlex class and nine in the Simion class. We also consider a refined face countwhere we distinguish between forward and backward arrows. We prove the refinedface counts only depend on the class of the triangulations. The refined face generatingfunctions are expressed in terms of the Catalan and Delannoy generatingfunctions and the modified Bessel function of the first kind.