On flow polytopes, order polytopes, and certain faces of the alternating sign matrix polytope

In this paper we study an alternating sign matrix analogue of the Chan-Robbins-Yuen polytope, which we call the ASM-CRY polytope. We show that this polytope has Catalan many vertices and its volume is equal to the number of standard Young tableaux of staircase shape; we also determine its Ehrhart polynomial. We achieve the previous by proving that the members of a family of faces of the alternating sign matrix polytope which includes ASM-CRY are both order and flow polytopes. Inspired by the above results, we relate three established triangulations of order and flow polytopes, namely Stanley's triangulation of order polytopes, the Postnikov-Stanley triangulation of flow polytopes and the Danilov-Karzanov-Koshevoy triangulation of flow polytopes. We show that when a graph $G$ is a planar graph, in which case the flow polytope $F_G$ is also an order polytope, Stanley's triangulation of this order polytope is one of the Danilov-Karzanov-Koshevoy triangulations of $F_G$. Moreover, for a general graph $G$ we show that the set of Danilov-Karzanov-Koshevoy triangulations of $F_G$ is a subset of the set of Postnikov-Stanley triangulations of $F_G$. We also describe explicit bijections between the combinatorial objects labeling the simplices in the above triangulations.