## Abstract

We give a combinatorial interpretation of the Lidskii formula for flow polytopes and use it to compute volumes via the enumeration of new families of combinatorial objects which are generalizations of parking functions. Our model applies to recover formulas of Pitman and Stanley, and compute volumes of previously seemingly unapproachable flow polytopes. A highlight of our model is that it leads to a combinatorial proof of an elegant volume formula for a new flow polytope which we call the caracol polytope. We prove that the volume of this polytope is the product of a Catalan number and the number of parking functions.

Original language | English |
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State | Published - 2018 |

Event | 30th international conference on Formal Power Series and Algebraic Combinatorics, FPSAC 2018 - Hanover, United States Duration: Jul 16 2018 → Jul 20 2018 |

### Conference

Conference | 30th international conference on Formal Power Series and Algebraic Combinatorics, FPSAC 2018 |
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Country/Territory | United States |

City | Hanover |

Period | 7/16/18 → 7/20/18 |

### Bibliographical note

Funding Information:∗c.benedetti@uniandes.edu.co. C. Benedetti thanks the Faculty of Science of Universidad de los Andes, York University, and Fields Institute for their support. †rafael.gonzalezl@usa.edu.co ‡chanusa@qc.cuny.edu. C. R. H. Hanusa is grateful for the support of PSC-CUNY Award 69120-0047. §peh2@williams.edu. P. E. Harris was supported by NSF award DMS-1620202. ¶khare@iisc.ac.in. A. Khare was partially supported by Ramanujan Fellowship SB/S2/RJN-121/2017 and MATRICS grant MTR/2017/000295 from SERB (Govt. of India), by grant F.510/25/CAS-II/2018(SAP-I) from UGC (Govt. of India), and by a Young Investigator Award from the Infosys Foundation. ‖ahmorales@math.umass.edu. A. H. Morales was partially supported by an AMS-Simons travel grant. ∗∗martha.yip@uky.edu. M. Yip was partially supported by Simons collaboration grant 429920.

Funding Information:

C. R. H. Hanusa is grateful for the support of PSC-CUNY Award 69120-0047. P. E. Harris was supported by NSF award DMS-1620202. A. Khare was partially supported by Ramanujan Fellowship SB/S2/RJN-121/2017 and MATRICS grant MTR/2017/000295 from SERB (Govt. of India), by grant F.510/25/CAS-II/2018(SAPI) from UGC (Govt. of India), and by a Young Investigator Award from the Infosys Foundation. A. H. Morales was partially supported by an AMS-Simons travel grant. M. Yip was partially supported by Simons collaboration grant 429920. Verma modules in representation theory, special functions, and algebraic combinatorics. Their combinatorial and geometric study started with work of Baldoni and Vergne [1] and unpublished work of Postnikov and Stanley. This project was initiated at the Polyhedral Geometry and Partition Theory workshop at the American Institute of Mathematics (AIM) in November 2016. We are extremely grateful to the organizers of the workshop - Federico Ardila, Benjamin Braun, Peter Paule, and Carla Savage - and to AIM which led to this collaboration.

Publisher Copyright:

© FPSAC 2018 - 30th international conference on Formal Power Series and Algebraic Combinatorics. All rights reserved.

## Keywords

- Catalan number
- Flow polytope
- Kostant partition function
- Labeled Dyck path
- Lidskii formula
- Parking function

## ASJC Scopus subject areas

- Algebra and Number Theory