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Product formulas for volumes of flow polytopes
- Publication Year :
- 2011
-
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 \prod_{i=m+1}^{m+n-2}\frac{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, \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.<br />Comment: 15 pages, 5 figures
- Subjects :
- Mathematics - Combinatorics
52B11, 05E99
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.1111.5634
- Document Type :
- Working Paper