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

Flow polytopes are an important class of polytopes in combinatorics whose lattice points and volumes have interesting properties and relations. The Chan–Robbins–Yuen (CRY) polytope is a flow polytope with normalized volume equal to the product of consecutive Catalan numbers. Zeilberger proved this by evaluating the Morris constant term identity, but no combinatorial proof is known. There is a refinement of this formula that splits the largest Catalan number into Narayana numbers, which Mészáros gave an interpretation as the volume of a collection of flow polytopes. We introduce a new refinement of the Morris identity with combinatorial interpretations both in terms of lattice points and volumes of flow polytopes. Our results generalize Mészáros’s construction and a recent flow polytope interpretation of the Morris identity by Corteel–Kim–Mészáros. We prove the product formula of our refinement following the strategy of the Baldoni–Vergne proof of the Morris identity. Lastly, we study a symmetry of the Morris identity bijectively using the Danilov–Karzanov–Koshevoy triangulation of flow polytopes and a bijection of Mészáros–Morales– Striker. Funding. This research was made possible by MIT PRIMES-USA 2020 program. Alejandro Morales was partially supported by the NSF Grant DMS-1855536. Manuscript received 9th March 2021, revised and accepted 26th April 2021.