Flow polytopes with Catalan volumes

The Chan–Robbins–Yuen polytope can be thought of as the flow polytope of the complete graph with netflow vector (1,0,…,0,−1). The normalized volume of the Chan–Robbins–Yuen polytope equals the product of consecutive Catalan numbers, yet there is no combinatorial proof of this fact. We consider a natural generalization of this polytope, namely, the flow polytope of the complete graph with netflow vector (1,1,0,…,0,−2). We show that the volume of this polytope is a certain power of 2 times the product of consecutive Catalan numbers. Our proof uses constant-term identities and further deepens the combinatorial mystery of why these numbers appear. In addition, we introduce two more families of flow polytopes whose volumes are given by product formulas.