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Flow polytopes of partitions. (English) Zbl 1409.52013
Summary: Recent progress on flow polytopes indicates many interesting families with product formulas for their volume. These product formulas are all proved using analytic techniques. Our work breaks from this pattern. We define a family of closely related flow polytopes \(\mathcal F_{(\lambda,\mathbf{a})}\) for each partition shape \(\lambda\) and netflow vector \(\mathbf{a}\in \mathbb{Z}^n_{>0}\). In each such family, we prove that there is a polytope (the limiting one in a sense) which is a product of scaled simplices, explaining their product volumes. We also show that the combinatorial type of all polytopes in a fixed family \(\mathcal{F}_{(\lambda,\mathbf{a})}\) is the same. When \(\lambda\) is a staircase shape and \(\mathbf{a}\) is the all ones vector the latter results specializes to a theorem of the first author with Morales and Rhoades [K. Mészáros et al., Sel. Math., New Ser. 23, No. 1, 425–454 (2017; Zbl 1355.05271)], which shows that the combinatorial type of the Tesler polytope is a product of simplices.

52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)
05C21 Flows in graphs
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