Publication Date

2021

Journal or Book Title

Comptes Rendus. Mathématique

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 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.

ORCID

https://orcid.org/0000-0002-2848-3230

DOI

https://doi.org/10.5802/crmath.218

Pages

823-851

Issue

7

License

UMass Amherst Open Access Policy

Creative Commons License

Creative Commons Attribution 4.0 License
This work is licensed under a Creative Commons Attribution 4.0 License.

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