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A combinatorial model for computing volumes of flow polytopes. (English) Zbl 1420.05011
Summary: We introduce new families of combinatorial objects whose enumeration computes volumes of flow polytopes. These objects provide an interpretation, based on parking functions, of W. Baldoni and M. Vergne’s generalization of a volume formula [Transform. Groups 13, No. 3–4, 447–469 (2008; Zbl 1200.52008)] 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.

##### MSC:
 05A15 Exact enumeration problems, generating functions 05A19 Combinatorial identities, bijective combinatorics 52B05 Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.) 52A38 Length, area, volume and convex sets (aspects of convex geometry) 05C20 Directed graphs (digraphs), tournaments 05C21 Flows in graphs 52B11 $$n$$-dimensional polytopes
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