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Experimental Mathematics Seminar

Refinements and symmetries for volumes of flow polytopes

Alejandro Morales, UMass, Amherst, and William Shi, Northview highschool, Johns Creek, GA.

Location:  Zoom
Date & time: Thursday, 25 March 2021 at 5:00PM - 6:00PM

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.
[password: The 20th Catalan number, alias (40)!/(20!*21!), alias 6564120420 ]

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