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Flow polytopes with Catalan volumes. (Polytopes de flot avec volumes de Catalan.) (English. French summary) Zbl 1358.05129
Summary: The Chan-Robbins-Yuen polytope can be thought of as the flow polytope of the complete graph with netflow vector \((1, 0, \ldots, 0, - 1)\). The normalized volume of the Chan-Robbins-Yuen polytope equals the product of consecutive Catalan numbers, yet there is no combinatorial proof of this fact. We consider a natural generalization of this polytope, namely, the flow polytope of the complete graph with netflow vector \((1, 1, 0, \ldots, 0, - 2)\). We show that the volume of this polytope is a certain power of 2 times the product of consecutive Catalan numbers. Our proof uses constant-term identities and further deepens the combinatorial mystery of why these numbers appear. In addition, we introduce two more families of flow polytopes whose volumes are given by product formulas.

05C21 Flows in graphs
05C22 Signed and weighted graphs
05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
51M25 Length, area and volume in real or complex geometry
Full Text: DOI
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