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Product formulas for volumes of flow polytopes. (English) Zbl 1310.51024
Summary: Intrigued by the product formula $$\prod _{i=1}^{n-2} C_i$$ for the volume of the Chan-Robbins-Yuen polytope $$\mathrm{CRY}_n$$, where $$C_i$$ is the $$i^{th}$$ Catalan number, we construct a family of polytopes $$\mathcal {P}_{m,n}$$, indexed by $$m \in \mathbb{Z}_{\geq 0}$$ and $$n \in \mathbb{Z}_{\geq 2}$$, whose volumes are given by the product $\prod _{i=m+1}^{m+n-1} \frac{1}{2i+1} {{m+n+i+1} \choose 2i} .$ The Chan-Robbins-Yuen polytope $$\mathrm{CRY}_n$$ coincides with $$\mathcal {P}_{0,n-2}$$. Our construction of the polytopes $$\mathcal {P}_{m,n}$$ is an application of a systematic method we develop for expressing volumes of a class of flow polytopes as the number of certain triangular arrays. This method can also be used as a heuristic technique for constructing polytopes with combinatorial volumes. As an illustration of this we construct polytopes whose volumes equal the number of $$r$$-ary trees on $$n$$ internal nodes, $$\frac {1}{(r-1)n+1} {{rn} \choose n}$$. Using triangular arrays we also express the volumes of flow polytopes as constant terms of formal Laurent series.

##### MSC:
 51M25 Length, area and volume in real or complex geometry 51M20 Polyhedra and polytopes; regular figures, division of spaces 05E10 Combinatorial aspects of representation theory 52B11 $$n$$-dimensional polytopes
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