On Product Formulas for Volumes of Flow Polytopes

Abstract

We outline the construction of 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 $$\displaystyle{\prod _{i=m+1}^{m+n-2} \frac{1} {2i + 1}{m + n + i\choose 2i}.}$$ The Chan-Robbins-Yuen polytope CRY n , whose volume is \(\prod _{i=1}^{n-2}C_{i}\), coincides with \(\mathcal{P}_{0,n-1}\). 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. The latter is also the constant term of a formal Laurent series.