A combinatorial model for computing volumes of 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. 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.Dedicated to the memory of Griff L. Bilbro.

On Volume Functions of Special Flow Polytopes Associated to the Root System of Type $A$

“…Hence the solution of ( 5) is unique up to a constant multiple. On the other hand, by Theorem 15, v A + r ,m,c nice satisfies the system of differential equations (5). Hence ϕ r is equal to a constant multiple of v A + r ,m,c nice .…”

“…We also refer to [1] for similar formulas for other chambers in more general settings. Moreover, we mention that a generalization of the Lidskii formula is shown in [3,Theorem 38], there is a geometric proof of the Lidskii formula in [12], and combinatorial applications of this formula are given in [5,7].…”

Refinements and Symmetries of the Morris identity for volumes of flow polytopes

“…Given that a combinatorial proof of the Morris identity has been elusive and would serve immediately as a combinatorial proof for the volume formula of the Chan-Robbins-Yuen polytope. See [5,16,32] for combinatorial proofs of volumes of flow polytopes F G for other graphs G.…”

Volumes of Flow Polytopes Related to Caracol Graphs

“…Since the discovery of the Chan-Robbins-Yuen polytope, researchers have found many flow polytopes whose volumes have nice product formulas, see [3,5,6,8,9,10,14] and references therein. In this paper we add another flow polytope to this list by proving a product formula for the volume of the flow polytope coming from a caracol graph, which was recently conjectured by Benedetti et al [3]. In order to state our results we introduce necessary definitions.…”

“…In [3], Benedetti et al introduced combinatorial models called gravity diagrams and unified diagrams to compute volumes of flow polytopes. Using these models they showed…”