Product formulas for volumes of flow polytopes

TL;DR

This paper introduces a systematic method to compute volumes of flow polytopes, generalizing known product formulas and connecting volumes to combinatorial objects like Catalan numbers and r-ary trees.

Contribution

The authors develop a new method to express flow polytope volumes via triangular arrays, enabling the construction of polytopes with volumes linked to various combinatorial counts.

Findings

Derived a product formula for volumes of a new family of polytopes P_{m,n}

Connected polytope volumes to Catalan numbers and r-ary trees

Expressed volumes as constant terms of Laurent series

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 ith Catalan number, we construct a family of polytopes P_{m,n}, whose volumes are given by the product \prod_{i=m+1}^{m+n-2}\frac{1}{2i+1}{{m+n+i} \choose {2i}}. The Chan-Robbins-Yuen polytope CRY_n coincides with P_{0,n-1}. Our construction of the polytopes 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.