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.
Categories: math.CO