Refinements and Symmetries of the Morris identity for volumes of flow polytopes

Flow polytopes are an important class of in combinatorics whose lattice points and volumes have interesting properties relations. The Chan-Robbins-Yuen (CRY) polytope is a flow with normalized volume equal to the product consecutive Catalan numbers. Zeilberger proved this by evaluating Morris constant term identity, but no combinatorial proof known. There refinement formula that splits largest number into Narayana numbers, which M\'esz\'aros gave interpretation as collection polytopes. We introduce new identity interpretations both terms Our results generalize M\'esz\'aros's construction recent Corteel-Kim-M\'esz\'aros. prove our following strategy Baldoni-Vergne identity. Lastly, we study symmetry bijectively using Danilov-Karzanov-Koshevoy triangulation bijection M\'esz\'aros-Morales-Striker.