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

TL;DR

This paper introduces a new combinatorial refinement of the Morris identity related to flow polytopes, generalizing previous interpretations and proving a product formula using established combinatorial and geometric methods.

Contribution

It presents a novel combinatorial refinement of the Morris identity with interpretations in lattice points and volumes, extending prior work and proving a new product formula.

Findings

New combinatorial refinement of Morris identity

Interpretations in terms of lattice points and flow polytope volumes

Proof of the product formula using Baldoni-Vergne strategy

Abstract

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 M\'esz\'aros 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 M\'esz\'aros's construction and a recent flow polytope interpretation of the Morris identity by Corteel-Kim-M\'esz\'aros. 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 M\'esz\'aros-Morales-Striker.