A combinatorial model for computing volumes of flow polytopes

Benedetti, Carolina; González D’León, Rafael; Hanusa, Christopher; Harris, Pamela; Khare, Apoorva; Morales, Alejandro; Yip, Martha

doi:10.1090/tran/7743

A combinatorial model for computing volumes of flow polytopes
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by Carolina Benedetti, Rafael S. González D’León, Christopher R. H. Hanusa, Pamela E. Harris, Apoorva Khare, Alejandro H. Morales and Martha Yip PDF

Trans. Amer. Math. Soc. 372 (2019), 3369-3404 Request permission

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.

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A combinatorial model for computing volumes of flow polytopes
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Abstract: