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