## 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

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