Product formulas for volumes of flow polytopes

2011/11/23 by Karola Mészáros, Meszaros, Karola
Agricultural and Biological Sciences · #05E99 #52B11 #Botanical Research and Chemistry #Combinatorics (math.CO) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.1111.5634

openalex publication_date 2011/11/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 ∏_i=m+1^m+n-2(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, (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.

Related

• Flow polytopes with Catalan volumes
• Volumes of generalized Chan-Robbins-Yuen polytopes
• Flow polytopes of partitions
• Volumes and Ehrhart polynomials of flow polytopes
• Counting integer points of flow polytopes