On Flow Polytopes, Order Polytopes, and Certain Faces of the Alternating Sign Matrix Polytope

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 tableaus 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 $${{\mathcal {F}}}_G$$FG is also an order polytope, Stanley’s triangulation of this order polytope is one of the Danilov–Karzanov–Koshevoy triangulations of $${{\mathcal {F}}}_G$$FG. Moreover, for a general graph G we show that the set of Danilov–Karzanov–Koshevoy triangulations of $${{\mathcal {F}}}_G$$FG equals the set of framed Postnikov–Stanley triangulations of $${{\mathcal {F}}}_G$$FG. We also describe explicit bijections between the combinatorial objects labeling the simplices in the above triangulations.