A combinatorial model for computing volumes of flow polytopes

TL;DR

This paper introduces a combinatorial framework using parking functions to compute flow polytope volumes, recovering known formulas and deriving new ones, including an elegant formula for the caracol graph's flow polytope.

Contribution

It presents a novel combinatorial model for flow polytope volumes, generalizes existing formulas, and introduces new sequences and conjectures related to these volumes.

Findings

Derived a new combinatorial interpretation for flow polytope volumes

Proved log-concavity of certain volume-related sequences

Introduced a new Ehrhart-like polynomial for flow polytopes

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 that were seemingly unapproachable. 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.