Flow polytopes with Catalan volumes

Corteel, Sylvie; Kim, Jang Soo; Mészáros, Karola

Comptes Rendus. Mathématique, Tome 355 (2017) no. 3, pp. 248-259.

Le polytope de Chan–Robbins–Yuen peut être considéré comme le polytope de flot du graphe complet avec vecteur de flot (1,0,…,0,−1). Le volume normalisé du polytope de Chan–Robbins–Yuen est égal au produit de nombres de Catalan consécutifs, mais il n'existe pas de preuve combinatoire de ce fait. Nous considérons une extension naturelle de ce polytope, à savoir le polytope de flot du graphe complet avec vecteur de flot (1,1,0,…,0,−2). Nous montrons que le volume de ce polytope est une certaine puissance de 2 fois le produit de nombres de Catalan consécutifs. Notre preuve utilise des identités de termes constants et approfondit encore le mystère combinatoire de la raison pour laquelle ces nombres apparaissent. De plus, nous introduisons deux familles de polytopes de flot dont les volumes sont donnés par des formules produits.

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.

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