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20
The Cayley Trick and triangulations of products of simplices
 in “Proceedings of the Joint Summer Research Conference on Integer Points in Polyhedra. Geometry, Number Theory, Algebra, and Optimization
"... Abstract. We use the Cayley Trick to study polyhedral subdivisions of the product of two simplices. For arbitrary (fixed) l ≥ 2, we show that the numbers of regular and nonregular triangulations of ∆ l × ∆ k grow, respectively, as k Θ(k) and 2 Ω(k2). For the special case of ∆ 2 × ∆ k, we relate t ..."
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Cited by 33 (3 self)
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Abstract. We use the Cayley Trick to study polyhedral subdivisions of the product of two simplices. For arbitrary (fixed) l ≥ 2, we show that the numbers of regular and nonregular triangulations of ∆ l × ∆ k grow, respectively, as k Θ(k) and 2 Ω(k2). For the special case of ∆ 2 × ∆ k, we relate triangulations to certain class of lozenge tilings. This allows us to compute the exact number of triangulations up to k = 15, show that the number grows as e βk2 /2+o(k 2) where β ≃ 0.32309594 and prove that the set of all triangulations is connected under geometric bistellar flips. The latter has as a corollary that the toric Hilbert scheme of the determinantal ideal of 2×2 minors of a 3×k matrix is connected, for every k. We include “Cayley Trick pictures ” of all the triangulations of ∆ 2 × ∆ 2 and ∆ 2 × ∆ 3, as well as one nonregular triangulation of ∆ 2 × ∆ 5 and another of ∆ 3 × ∆ 3.
Triangulations of cyclic polytopes and higher Bruhat orders
 MATHEMATIKA
, 1997
"... Recently EDELMAN & REINER suggested two poset structures S 1 (n;d) and S 2 (n;d) on the set of all triangulations of the cyclic dpolytope C(n;d) with n vertices. Both posets are generalizations of the wellstudied Tamari lattice. While S 2 (n;d) is bounded by definition, the same is not obvio ..."
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Cited by 33 (5 self)
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Recently EDELMAN & REINER suggested two poset structures S 1 (n;d) and S 2 (n;d) on the set of all triangulations of the cyclic dpolytope C(n;d) with n vertices. Both posets are generalizations of the wellstudied Tamari lattice. While S 2 (n;d) is bounded by definition, the same is not obvious for S 1 (n;d). In the paper by EDELMAN & REINER the bounds of S 2 (n;d) were also confirmed for S 1 (n;d) whenever d 5, leaving the general case as a conjecture. In this paper their conjecture is answered in the affirmative for all d, using several new functorial constructions. Moreover, a structure theorem is presented, stating that the elements of S 1 (n;d + 1) are in onetoone correspondence to certain equivalence classes of maximal chains in S 1 (n;d). By similar methods it is proved that all triangulations of cyclic polytopes are shellable. In order to clarify the connection between S 1 (n;d) and the higher Bruhat order B(n \Gamma 2;d \Gamma 1) of MANIN & SCHECHTMAN, we define an orderpreserving map from B(n \Gamma 2;d \Gamma 1) to S 1 (n;d), thereby concretizing a result by KAPRANOV & VOEVODSKY in the theory of ordered ncategories.
Tropical hyperplane arrangements and oriented matroids
 Math. Zeitschrift
"... Abstract. We study the combinatorial properties of a tropical hyperplane arrangement. We define tropical oriented matroids, and prove that they share many of the properties of ordinary oriented matroids. We show that a tropical oriented matroid determines a subdivision of a product of two simplices, ..."
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Cited by 27 (1 self)
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Abstract. We study the combinatorial properties of a tropical hyperplane arrangement. We define tropical oriented matroids, and prove that they share many of the properties of ordinary oriented matroids. We show that a tropical oriented matroid determines a subdivision of a product of two simplices, and conjecture that this correspondence is a bijection. 1.
Triangulations Of Oriented Matroids
, 1997
"... We consider the concept of triangulation of an oriented matroid. We provide a definition which generalizes the previous ones by Billera Munson and by Anderson and which specializes to the usual notion of triangulation (or simplicial fan) in the realizable case. Then we study the relation existing ..."
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Cited by 26 (10 self)
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We consider the concept of triangulation of an oriented matroid. We provide a definition which generalizes the previous ones by Billera Munson and by Anderson and which specializes to the usual notion of triangulation (or simplicial fan) in the realizable case. Then we study the relation existing between triangulations of an oriented matroid M and extensions of its dual M , via the socalled lifting triangulations. We show that this duality behaves particularly well in the class of Lawrence matroid polytopes. In particular, that the extension space conjecture for realizable oriented matroids posed by Sturmfels and Ziegler is equivalent to the restriction to Lawrence polytopes of the Generalized Baues problem for subdivisions of polytopes. We finish showing examples and a combinatorial characterization of lifting triangulations. Introduction Matroids (see [23]) and oriented matroids (see [8]) are axiomatic abstract models for combinatorial geometry over general fields and ordere...
Flag arrangements and triangulations of products of simplices
 FORMAL POWER SERIES AND ALGEBRAIC COMBINATORICS
, 2006
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Products of foldable triangulations
, 2006
"... Regular triangulations of products of lattice polytopes are constructed with the additional property that the dual graphs of the triangulations are bipartite. The (weighted) size difference of this bipartition is a lower bound for the number of real roots of certain sparse polynomial systems by rece ..."
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Cited by 10 (2 self)
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Regular triangulations of products of lattice polytopes are constructed with the additional property that the dual graphs of the triangulations are bipartite. The (weighted) size difference of this bipartition is a lower bound for the number of real roots of certain sparse polynomial systems by recent results of Soprunova and Sottile [Adv. Math. 204(1):116–151, 2006]. Special attention is paid to the cube case.
Combinatorics with a geometric flavor: some examples
 in Visions in Mathematics Toward 2000 (Geometric and Functional Analysis, Special Volume
, 2000
"... In this paper I try to present my field, combinatorics, via five examples of combinatorial studies which have some geometric flavor. The first topic is Tverberg's theorem, a gem in combinatorial geometry, and various of its combinatorial and topological extensions. McMullen's upper bound t ..."
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Cited by 8 (0 self)
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In this paper I try to present my field, combinatorics, via five examples of combinatorial studies which have some geometric flavor. The first topic is Tverberg's theorem, a gem in combinatorial geometry, and various of its combinatorial and topological extensions. McMullen's upper bound theorem for the face numbers of convex polytopes and its many extensions is the second topic. Next are general properties of subsets of the vertices of the discrete ndimensional cube and some relations with questions of extremal and probabilistic combinatorics. Our fourth topic is tree enumeration and random spanning trees, and finally, some combinatorial and geometrical aspects of the simplex method for linear programming are considered.
Polytope Skeletons And Paths
 HANDBOOK OF DISCRETE AND COMPUTATIONAL GEOMETRY (SECOND EDITION ), CHAPTER 20
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Shelling and the hVector of the (extra)ordinary Polytope
, 2005
"... Ordinary polytopes were introduced by Bisztriczky as a (nonsimplicial) generalization of cyclic polytopes. We show that the colex order of facets of the ordinary polytope is a shelling order. This shelling shares many nice properties with the shellings of simplicial polytopes. We also give a shallo ..."
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Cited by 4 (2 self)
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Ordinary polytopes were introduced by Bisztriczky as a (nonsimplicial) generalization of cyclic polytopes. We show that the colex order of facets of the ordinary polytope is a shelling order. This shelling shares many nice properties with the shellings of simplicial polytopes. We also give a shallow triangulation of the ordinary polytope, and show how the shelling and the triangulation are used to compute the toric hvector of the ordinary polytope. As one consequence, we get that the contribution from each shelling component to the hvector is nonnegative. Another consequence is a combinatorial proof that the entries of the hvector of any ordinary polytope are simple sums of binomial coefficients.
Triangulations of ∆n−1 × ∆d−1 and Tropical Oriented Matroids
"... Abstract. Develin and Sturmfels showed that regular triangulations of ∆n−1 × ∆d−1 can be thought of as tropical polytopes. Tropical oriented matroids were defined by Ardila and Develin, and were conjectured to be in bijection with all subdivisions of ∆n−1 × ∆d−1. In this paper, we show that any tria ..."
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Cited by 4 (1 self)
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Abstract. Develin and Sturmfels showed that regular triangulations of ∆n−1 × ∆d−1 can be thought of as tropical polytopes. Tropical oriented matroids were defined by Ardila and Develin, and were conjectured to be in bijection with all subdivisions of ∆n−1 × ∆d−1. In this paper, we show that any triangulation of ∆n−1 × ∆d−1 encodes a tropical oriented matroid. We also suggest a new class of combinatorial objects that may describe all subdivisions of a bigger class of polytopes. Résumé. Develin et Sturmfels ont montré que les triangulations de ∆n−1 × ∆d−1 peuvent être considérées comme des polytopes tropicaux. Les matroïdes orientés tropicaux ont été définis par Ardila et Develin, et ils ont été conjecturés être en bijection avec les subdivisions de ∆n−1 ×∆d−1. Dans cet article, nous montrons que toute triangulation de ∆n−1 × ∆d−1 encode un matroïde orienté tropical. De plus, nous proposons une nouvelle classe d’objets combinatoires qui peuvent décrire toutes les subdivisions d’une plus grande classe de polytopes.