## A combinatorial model for computing volumes of flow polytopes

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- by Carolina Benedetti, Rafael S. González D’León, Christopher R. H. Hanusa, Pamela E. Harris, Apoorva Khare, Alejandro H. Morales and Martha Yip PDF
- Trans. Amer. Math. Soc.
**372**(2019), 3369-3404 Request permission

## Abstract:

We introduce new families of combinatorial objects whose enumeration computes volumes of flow polytopes. These objects provide an interpretation, based on parking functions, of Baldoni and Vergne’s generalization of a volume formula originally due to Lidskii. We recover known flow polytope volume formulas and prove new volume formulas for flow polytopes. A highlight of our model is an elegant formula for the flow polytope of a graph we call the caracol graph.

As by-products of our work, we uncover a new triangle of numbers that interpolates between Catalan numbers and the number of parking functions, we prove the log-concavity of rows of this triangle along with other sequences derived from volume computations, and we introduce a new Ehrhart-like polynomial for flow polytope volume and conjecture product formulas for the polytopes we consider.

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## Additional Information

**Carolina Benedetti**- Affiliation: Departamento de Matemáticas, Universidad de los Andes, Bogotá, Colombia
- MR Author ID: 898089
- Email: c.benedetti@uniandes.edu.co
**Rafael S. González D’León**- Affiliation: Escuela de Ciencias Exactas e Ingeniería, Universidad Sergio Arboleda, Bogotá, Colombia
- Email: rafael.gonzalezl@usa.edu.co
**Christopher R. H. Hanusa**- Affiliation: Department of Mathematics, Queens College (CUNY), 65-30 Kissena Boulevard, Flushing, New York 11367
- MR Author ID: 723495
- Email: chanusa@qc.cuny.edu
**Pamela E. Harris**- Affiliation: Department of Mathematics and Statistics, Williams College, Bascom House, Room 106C, 33 Stetson Court, Williamstown, Massachusetts 01267
- MR Author ID: 953833
- ORCID: 0000-0002-3049-991X
- Email: peh2@williams.edu
**Apoorva Khare**- Affiliation: Department of Mathematics, Indian Institute of Science, Analysis and Probability Research Group, Bangalore 560012, India
- MR Author ID: 750359
- ORCID: 0000-0002-1577-9171
- Email: khare@iisc.ac.in
**Alejandro H. Morales**- Affiliation: Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003
- MR Author ID: 819004
- Email: ahmorales@math.umass.edu
**Martha Yip**- Affiliation: Department of Mathematics, University of Kentucky, 715 Patterson Office Tower, Lexington, Kentucky 40506-0027
- MR Author ID: 805658
- Email: martha.yip@uky.edu
- Received by editor(s): January 30, 2018
- Received by editor(s) in revised form: October 26, 2018
- Published electronically: May 23, 2019
- Additional Notes: The first author was supported by FAPA grant from Universidad de los Andes Faculty of Science, and by York University and the Fields Institute

The second author was supported during this project by the University of Kentucky, York University, and Universidad Sergio Arboleda, and he is grateful for their support.

The third author is grateful for the support of PSC-CUNY Award 69120-0047.

The fourth author was supported by NSF award DMS-1620202.

The fifth author was partially supported by Ramanujan Fellowship SB/S2/RJN-121/2017 and MATRICS grant MTR/2017/000295 from SERB (Government of India), by grant F.510/25/CAS-II/2018(SAP-I) from UGC (Government of India), and by a Young Investigator Award from the Infosys Foundation.

The sixth author was partially supported by an AMS-Simons Travel Grant.

The seventh author was partially supported by Simons Collaboration Grant 429920. - © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**372**(2019), 3369-3404 - MSC (2010): Primary 05A15, 05A19, 52B05, 52A38; Secondary 05C20, 05C21, 52B11
- DOI: https://doi.org/10.1090/tran/7743
- MathSciNet review: 3988614

Dedicated: Dedicated to the memory of Griff L. Bilbro