A combinatorial model for computing volumes of flow polytopes

Article with Carolina Benedetti, Rafael Gonzalez D’Leon, Christopher R. H. Hanusa, Apoorva Khare, Alejandro Morales and Martha Yip.

Abstract

We introduce new families of combinatorial objects whose enumeration com- putes 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 Lid- skii. 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.