## 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.

Original language | English |
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Pages (from-to) | 3369-3404 |

Number of pages | 36 |

Journal | Transactions of the American Mathematical Society |

Volume | 372 |

Issue number | 5 |

DOIs | |

State | Published - Sep 1 2019 |

### Bibliographical note

Publisher Copyright:© 2019 American Mathematical Society.

## Keywords

- Binomial transform
- Caracol graph
- Catalan numbers
- Chan–Robbins–Yuen polytope
- Dyck path
- Ehrhart polynomial
- Flow polytope
- Gravity diagram
- Kostant partition function
- Lidskii formula
- Line-dot diagram
- Log-concave
- Multi-labeled Dyck path
- Parking function
- Parking triangle
- Pitman–Stanley polytope
- Tesler polytope
- Unified diagram
- Zigzag graph

## ASJC Scopus subject areas

- General Mathematics
- Applied Mathematics