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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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