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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.

MSC:
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
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[1] Baldoni, W.; Vergne, M., Kostant partitions functions and flow polytopes, Transform. Groups, 13, 3-4, 447-469, (2008) · Zbl 1200.52008
[2] Chan, C. S.; Robbins, D. P., On the volume of the polytope of doubly stochastic matrices, Exp. Math., 8, 3, 291-300, (1999) · Zbl 0951.05015
[3] Chan, C. S.; Robbins, D. P.; Yuen, D. S., On the volume of a certain polytope, Exp. Math., 9, 1, 91-99, (2000) · Zbl 0960.05004
[4] Escobar, L.; Mészáros, K., Subword complexes via triangulations of root polytopes, (2015) · Zbl 1393.52010
[5] Escobar, L.; Mészáros, K., Toric matrix Schubert varieties and their polytopes, Proc. Amer. Math. Soc., 144, 12, 5081-5096, (2016) · Zbl 1357.14064
[6] L. Hille, Quivers, cones and polytopes, Linear Algebra Appl. (365), 215-237. · Zbl 1034.52011
[7] Liu, R. I.; Mészáros, K.; Morales, A. H., Flow polytopes and the space of diagonal harmonics, (2016)
[8] Mészáros, K., Product formulas for volumes of flow polytopes, Proc. Amer. Math. Soc., 3, 937-954, (2015) · Zbl 1310.51024
[9] Mészáros, K., Pipe dream complexes and triangulations of root polytopes belong together, SIAM J. Discrete Math., 30, 1, 100-111, (2016) · Zbl 1329.05317
[10] Mészáros, K.; Morales, A. H., Flow polytopes of signed graphs and the Kostant partition function, Int. Math. Res. Not., 3, 830-871, (2015) · Zbl 1307.05097
[11] Mészáros, K.; Morales, A. H.; Rhoades, B., The polytope of tesler matrices, Sel. Math., 23, 1, 425-454, (2017) · Zbl 1355.05271
[12] Morris, W. G., Constant term identities for finite and affine root systems: conjectures and theorems, (1982), University of Wisconsin-Madison, PhD Thesis
[13] Zeilberger, D., Proof of a conjecture of chan, robbins, and yuen, Electron. Trans. Numer. Anal., 9, 147-148, (1999) · Zbl 0941.05006
[14] Zeilberger, D., Sketch of a proof of an intriguing conjecture of karola meszaros and alejandro morales regarding the volume of the \(D_n\) analog of the chan-robbins-yuen polytope (or: the Morris-Selberg constant term identity strikes again!), (2014)
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