Product formulas for volumes of flow polytopes

Abstract:

Intrigued by the product formula $\prod _{i=1}^{n-2} C_i$ for the volume of the Chan-Robbins-Yuen polytope $CRY_n$, 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 $CRY_n$ 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.