3 research outputs found
A combinatorial model for computing volumes of flow polytopes
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.Comment: 34 pages, 15 figures. v2: updated after referee reports; includes a
proof of Proposition 8.7. Accepted into Transactions of the AM