Flow polytopes with Catalan volumes

Sylvie Corteel; Jang Soo Kim; Karola Mészáros

doi:10.1016/j.crma.2017.01.007

Combinatorics

Flow polytopes with Catalan volumes

Presented by: Michèle Vergne

Sylvie Corteel ¹; Jang Soo Kim ²; Karola Mészáros ³

¹ IRIF, CNRS et Université Paris-Diderot, 75205 Paris cedex 13, France
² Sungkyunkwan University, 2066 Seobu-ro, Jangan-gu, Suwon, Gyeonggi-do 16419, South Korea
³ Department of Mathematics, Cornell University, Ithaca NY 14853, USA

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

Abstract
Résumé

The Chan–Robbins–Yuen polytope can be thought of as the flow polytope of the complete graph with netflow vector $(1, 0, \dots, 0, - 1)$ . 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 $(1, 1, 0, \dots, 0, - 2)$ . 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.

Le polytope de Chan–Robbins–Yuen peut être considéré comme le polytope de flot du graphe complet avec vecteur de flot $(1, 0, \dots, 0, - 1)$ . 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 $(1, 1, 0, \dots, 0, - 2)$ . 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.

Received: 2016-12-22
Accepted: 2017-01-17
Published online: 2017-02-16

DOI: 10.1016/j.crma.2017.01.007

Author's affiliations:

Sylvie Corteel ¹; Jang Soo Kim ²; Karola Mészáros ³

¹ IRIF, CNRS et Université Paris-Diderot, 75205 Paris cedex 13, France
² Sungkyunkwan University, 2066 Seobu-ro, Jangan-gu, Suwon, Gyeonggi-do 16419, South Korea
³ Department of Mathematics, Cornell University, Ithaca NY 14853, USA

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     author = {Sylvie Corteel and Jang Soo Kim and Karola M\'esz\'aros},
     title = {Flow polytopes with {Catalan} volumes},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {248--259},
     publisher = {Elsevier},
     volume = {355},
     number = {3},
     year = {2017},
     doi = {10.1016/j.crma.2017.01.007},
     language = {en},
}

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Sylvie Corteel; Jang Soo Kim; Karola Mészáros. Flow polytopes with Catalan volumes. Comptes Rendus. Mathématique, Volume 355 (2017) no. 3, pp. 248-259. doi : 10.1016/j.crma.2017.01.007. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2017.01.007/

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Cited by Sources:

^☆ Corteel is partially supported by the project Emergences “Combinatoire à Paris”. Kim is partially supported by National Research Foundation of Korea (NRF) grants (NRF-2016R1D1A1A09917506) and (NRF-2016R1A5A1008055). Mészáros is partially supported by a National Science Foundation Grant (DMS 1501059).

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