Flow Polytopes of Partitions (original) (raw)
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Counting Integer Points of Flow Polytopes
Discrete & Computational Geometry, 2021
The Baldoni–Vergne volume and Ehrhart polynomial formulas for flow polytopes are significant in at least two ways. On one hand, these formulas are in terms of Kostant partition functions, connecting flow polytopes to this classical vector partition function, fundamental in representation theory. On the other hand, the Ehrhart polynomials can be read off from the volume functions of flow polytopes. The latter is remarkable since the leading term of the Ehrhart polynomial of an integer polytope is its volume! Baldoni and Vergne proved these formulas via residues. To reveal the geometry of these formulas, the second author and Morales gave a fully geometric proof for the volume formula and a partial generating function proof for the Ehrhart polynomial formula. The goal of the present paper is to provide a fully geometric proof for the Ehrhart polynomial formula for flow polytopes.
Flow Polytopes of Signed Graphs and the Kostant Partition Function
International Mathematics Research Notices, 2013
We establish the relationship between volumes of flow polytopes associated to signed graphs and the Kostant partition function. A special case of this relationship, namely, when the graphs are signless, has been studied in detail by Baldoni and Vergne using techniques of residues. In contrast with their approach, we provide entirely combinatorial proofs inspired by the work of Postnikov and Stanley on flow polytopes. As a fascinating special family of flow polytopes, we study the Chan-Robbins-Yuen polytopes. Motivated by the beautiful volume formula n−2 k=1 Cat(k) for the type An version, where Cat(k) is the kth Catalan number, we introduce type Cn+1 and Dn+1 Chan-Robbins-Yuen polytopes along with intriguing conjectures pertaining to their properties.
Flow polytopes and the graph of reflexive polytopes
Discrete Mathematics, 2009
We suggest defining the structure of an unoriented graph R d on the set of reflexive polytopes of a fixed dimension d. The edges are induced by easy mutations of the polytopes to create the possibility of walks along connected components inside this graph. For this, we consider two types of mutations: Those provided by performing duality via nef-partitions, and those arising from varying the lattice. Then for d ≤ 3, we identify the flow polytopes among the reflexive polytopes of each single component of the graph R d . For this, we present for any dimension d ≥ 2 an explicit finite list of quivers giving all d-dimensional reflexive flow polytopes up to lattice isomorphism. We deduce as an application that any such polytope has at most 6(d − 1) facets.
The polytope of Tesler matrices
Selecta Mathematica, 2016
We introduce the Tesler polytope Tesn(a), whose integer points are the Tesler matrices of size n with hook sums a 1 , a 2 , . . . , an ∈ Z ≥0 . We show that Tesn(a) is a flow polytope and therefore the number of Tesler matrices is counted by the type An Kostant partition function evaluated at (a 1 , a 2 , . . . , an, − n i=1 a i ). We describe the faces of this polytope in terms of "Tesler tableaux" and characterize when the polytope is simple. We prove that the h-vector of Tesn(a) when all a i > 0 is given by the Mahonian numbers and calculate the volume of Tesn(1, 1, . . . , 1) to be a product of consecutive Catalan numbers multiplied by the number of standard Young tableaux of staircase shape.
Representations and characterizations of vertices of bounded-shape partition polytopes
Linear Algebra and its Applications, 1998
Consider a finite set whose elements are associated with vectors of common dimension. A partition of such a set is associated with a matrix whose columns are the sums of the vectors corresponding to each part. The partition polytope associated with a class of partitions (that share the number of parts) is then the convex hull of the corresponding matrices. We derive representations and characterizations of these polytopes and their vertices. 0 1998 Elsevier Science Inc. All rights reserved.
On 2-Level Polytopes Arising in Combinatorial Settings
SIAM Journal on Discrete Mathematics
2-level polytopes naturally appear in several areas of pure and applied mathematics, including combinatorial optimization, polyhedral combinatorics, communication complexity, and statistics. In this paper, we present a study of some 2-level polytopes arising in combinatorial settings. Our first contribution is proving that f0(P)f d−1 (P) ≤ d2 d+1 for a large collection of families of such polytopes P. Here f0(P) (resp. f d−1 (P)) is the number of vertices (resp. facets) of P , and d is its dimension. Whether this holds for all 2-level polytopes was asked in [7], and experimental results from [16] showed it true for d ≤ 7. The key to most of our proofs is a deeper understanding of the relations among those polytopes and their underlying combinatorial structures. This leads to a number of results that we believe to be of independent interest: a trade-off formula for the number of cliques and stable sets in a graph; a description of stable matching polytopes as affine projections of certain order polytopes; and a linear-size description of the base polytope of matroids that are 2-level in terms of cuts of an associated tree.
The boundary volume of a lattice polytope
Bulletin of the Australian Mathematical Society, 2012
For a d-dimensional convex lattice polytope P, a formula for the boundary volume vol(∂P) is derived in terms of the number of boundary lattice points on the first [d/2] dilations of P . As an application we give a necessary and sufficient condition for a polytope to be reflexive, and derive formulae for the f-vector of a smooth polytope in dimensions 3, 4, and 5. We also give applications to reflexive order polytopes, and to the Birkhoff polytope.
Journal of Combinatorial Theory, Series A, 2006
We construct a new 2-parameter family E mn , m, n ≥ 3, of self-dual 2-simple and 2-simplicial 4-polytopes, with flexible geometric realisations. E 44 is the 24-cell. For large m, n the f -vectors have "fatness" close to 6.