zbMATH — the first resource for mathematics

Product formulas for volumes of flow polytopes. (English) Zbl 1310.51024
Summary: Intrigued by the product formula \( \prod _{i=1}^{n-2} C_i\) for the volume of the Chan-Robbins-Yuen polytope \(\mathrm{CRY}_n\), 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 \[ \prod _{i=m+1}^{m+n-1} \frac{1}{2i+1} {{m+n+i+1} \choose 2i} . \] The Chan-Robbins-Yuen polytope \(\mathrm{CRY}_n\) 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.

51M25 Length, area and volume in real or complex geometry
51M20 Polyhedra and polytopes; regular figures, division of spaces
05E10 Combinatorial aspects of representation theory
52B11 \(n\)-dimensional polytopes
Full Text: DOI
[1] [bv] Mich\`ele Vergne, Residue formulae for volumes and Ehrhart polynomials of convex polytopes, http://arxiv.org/abs/math/0103097.
[2] Baldoni, Welleda; Vergne, Mich{\`e}le, Kostant partitions functions and flow polytopes, Transform. Groups, 13, 3-4, 447-469 (2008) · Zbl 1200.52008
[3] Chan, Clara S.; Robbins, David P.; Yuen, David S., On the volume of a certain polytope, Experiment. Math., 9, 1, 91-99 (2000) · Zbl 0960.05004
[4] Kirillov, Anatol N., Ubiquity of Kostka polynomials. Physics and combinatorics 1999 (Nagoya), 85-200 (2001), World Sci. Publ., River Edge, NJ · Zbl 0982.05105
[5] Loehr, Nicholas A., Bijective combinatorics, Discrete Mathematics and its Applications (Boca Raton), xxii+590 pp. (2011), CRC Press: Boca Raton, FL:CRC Press · Zbl 1234.05001
[6] M{\'e}sz{\'a}ros, Karola, Root polytopes, triangulations, and the subdivision algebra. I, Trans. Amer. Math. Soc., 363, 8, 4359-4382 (2011) · Zbl 1233.05215
[7] M{\'e}sz{\'a}ros, Karola, Root polytopes, triangulations, and the subdivision algebra, II, Trans. Amer. Math. Soc., 363, 11, 6111-6141 (2011) · Zbl 1233.05216
[8] [mm] Karola M\'esz\'aros and Alejandro H. Morales, Flow polytopes and the Kostant partition function, http://arxiv.org/abs/1208.0140. · Zbl 1307.05097
[9] [wm] W. G. Morris, Constant Term Identities for Finite and Affine Root Systems: Conjectures and Theorems, Ph.D. Thesis, 1982.
[10] [p] A. Postnikov, personal communication, 2010.
[11] [S] Richard P. Stanley, Acyclic flow polytopes and Kostant’s partition function, Conference transparencies, 2000, http://math.mit.edu/\( \sim\) rstan/trans.html.
[12] Stanley, Richard P.; Pitman, Jim, A polytope related to empirical distributions, plane trees, parking functions, and the associahedron, Discrete Comput. Geom., 27, 4, 603-634 (2002) · Zbl 1012.52019
[13] [sta] Richard P. Stanley, personal communication, 2008.
[14] Zeilberger, Doron, Proof of a conjecture of Chan, Robbins, and Yuen, Orthogonal polynomials: numerical and symbolic algorithms (Legan\'es, 1998). Electron. Trans. Numer. Anal., 9, 147-148 (electronic) (1999) · Zbl 0941.05006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.