##### Department of Mathematics,

University of California San Diego

****************************

### Math 269 - Combinatorics Seminar

## Alejandro Morales

#### University of Massachusetts, Amherst

## Juggling, refinements and symmetries for volumes of flow polytopes

##### Abstract:

Flow polytopes are an important class of polytopes in combinatorics whose lattice points and volumes have interesting properties and relations to other parts of geometric and algebraic combinatorics. These polytopes were recently related to (multiplex) juggling sequences of Butler, Graham and Chung. 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 MÃ©szÃ¡ros gave an interpretation as the volume of a collection of flow polytopes. In this talk we will talk about the connection between juggling and flow polytopes and introduce a new refinement of the Morris identity with combinatorial interpretations both in terms of lattice points and volumes of flow polytopes. \\ \\ The first part of the talk is based on joint work with Benedetti, Hanusa, Harris and Simpson and the second part is based on joint work with William Shi.

Host: Brendon Rhoades

### April 9, 2021

### 1:00 PM

### Zoom ID: 966 3389 8784

****************************