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Product formulas for volumes of flow polytopes


Author: Karola Mészáros
Journal: Proc. Amer. Math. Soc. 143 (2015), 937-954
MSC (2010): Primary 05E10, 51M25, 52B11
DOI: https://doi.org/10.1090/S0002-9939-2014-12182-4
Published electronically: November 6, 2014
MathSciNet review: 3293712
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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 $ i^{th}$ Catalan number, we construct a family of polytopes $ \mathcal {P}_{m,n}$, indexed by $ m \in \mathbb{Z}_{\geq 0}$ and $ n \in \mathbb{Z}_{\geq 2}$, whose volumes are given by the product

$\displaystyle \prod _{i=m+1}^{m+n-1}\frac {1}{2i+1}{{m+n+i+1} \choose {2i}}.$

The Chan-Robbins-Yuen polytope $ CRY_n$ coincides with $ \mathcal {P}_{0,n-2}$. Our construction of the polytopes $ \mathcal {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.

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Additional Information

Karola Mészáros
Affiliation: Department of Mathematics, Cornell University, Ithaca, New York 14853

DOI: https://doi.org/10.1090/S0002-9939-2014-12182-4
Keywords: Flow polytope, Chan-Robbins-Yuen polytope, triangular arrays, triangulation, volume, Kostant partition function
Received by editor(s): December 1, 2011
Received by editor(s) in revised form: November 30, 2012, and February 23, 2013
Published electronically: November 6, 2014
Additional Notes: The author was supported by a National Science Foundation Postdoctoral Research Fellowship (DMS 1103933)
Communicated by: Jim Haglund
Article copyright: © Copyright 2014 Karola Mészáros