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arXiv · 1910.10060

A Fuss-Catalan variation of the caracol flow polytope

Martha Yip
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Abstract

Recently, a combinatorial interpretation of Baldoni and Vergne's generalized Lidskii formula for the volume of a flow polytope was developed by Benedetti et al.. This converts the problem of computing Kostant partition functions into a problem of enumerating a set of objects called unified diagrams. We devise an enhanced version of this combinatorial model to compute the volumes of flow polytopes defined on a family of graphs called the k-caracol graphs, resulting in the first application of the model to non-planar graphs. At k=1 and k=n-1, we recover results for the classical caracol graph and the Pitman--Stanley graph. Furthermore, we introduce the notion of in-degree gravity diagrams for flow polytopes, which are equinumerous with (out-degree) gravity diagrams considered by Benedetti et al.. We show that for the k-caracol flow polytopes, these two kinds of gravity diagrams satisfy a natural combinatorial correspondence, which raises an intriguing question on the relationship in the geometry of two related polytopes.

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Martha Yip. 2019-10-22. A Fuss-Catalan variation of the caracol flow polytope. https://arxiv.org/abs/1910.10060

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