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A REALIZATION OF GRAPHASSOCIAHEDRA
"... Abstract. Given any finite graph G, we offer a simple realization of the graphassociahedron PG using integer coordinates. 1. ..."
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Abstract. Given any finite graph G, we offer a simple realization of the graphassociahedron PG using integer coordinates. 1.
Multitriangulations as complexes of star polygons
, 2007
"... Maximal (k+1)crossingfree graphs on a planar point set in convex position, that is, ktriangulations, have received attention in recent literature, with motivation coming from several interpretations of them. We introduce a new way of looking at ktriangulations, namely as complexes of star poly ..."
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Cited by 23 (8 self)
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Maximal (k+1)crossingfree graphs on a planar point set in convex position, that is, ktriangulations, have received attention in recent literature, with motivation coming from several interpretations of them. We introduce a new way of looking at ktriangulations, namely as complexes of star polygons. With this tool we give new, direct, proofs of the fundamental properties of ktriangulations, as well as some new results. This interpretation also opensup new avenues of research, that we briefly explore in the last section.
PERMUTAHEDRA AND GENERALIZED ASSOCIAHEDRA
, 2008
"... Given a finite Coxeter system (W, S) and a Coxeter element c, or equivalently an orientation of the Coxeter graph of W, we construct a simple polytope whose outer normal fan is N. Reading’s Cambrian fan Fc, settling a conjecture of Reading that this is possible. We call this polytope the cgeneral ..."
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Given a finite Coxeter system (W, S) and a Coxeter element c, or equivalently an orientation of the Coxeter graph of W, we construct a simple polytope whose outer normal fan is N. Reading’s Cambrian fan Fc, settling a conjecture of Reading that this is possible. We call this polytope the cgeneralized associahedron. Our approach generalizes Loday’s realization of the associahedron (a type A cgeneralized associahedron whose outer normal fan is not the cluster fan but a coarsening of the Coxeter fan arising from the Tamari lattice) to any finite Coxeter group. A crucial role in the construction is played by the csingleton cones, the cones in the cCambrian fan which consist of a single maximal cone from the Coxeter fan. Moreover, if W is a Weyl group and the vertices of the permutahedron are chosen in a lattice associated to W, then we show that our realizations have
Multitriangulations, pseudotriangulations and primitive sorting networks
 Discrete Comput. Geom. (DOI
, 2012
"... Abstract. We study the set of all pseudoline arrangements with contact points which cover a given support. We define a natural notion of flip between these arrangements and study the graph of these flips. In particular, we provide an enumeration algorithm for arrangements with a given support, based ..."
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Abstract. We study the set of all pseudoline arrangements with contact points which cover a given support. We define a natural notion of flip between these arrangements and study the graph of these flips. In particular, we provide an enumeration algorithm for arrangements with a given support, based on the properties of certain greedy pseudoline arrangements and on their connection with sorting networks. Both the running time per arrangement and the working space of our algorithm are polynomial. As the motivation for this work, we provide in this paper a new interpretation of both pseudotriangulations and multitriangulations in terms of pseudoline arrangements on specific supports. This interpretation explains their common properties and leads to a natural definition of multipseudotriangulations, which generalizes both. We study elementary properties of multipseudotriangulations and compare them to iterations of pseudotriangulations.
BRICK POLYTOPES OF SPHERICAL SUBWORD COMPLEXES: A NEW APPROACH TO GENERALIZED ASSOCIAHEDRA
"... Abstract. We generalize the brick polytope of V. Pilaud and F. Santos to spherical subword complexes for finite Coxeter groups. This construction provides polytopal realizations for a certain class of subword complexes containing all cluster complexes of finite types. For the latter, the brick polyt ..."
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Abstract. We generalize the brick polytope of V. Pilaud and F. Santos to spherical subword complexes for finite Coxeter groups. This construction provides polytopal realizations for a certain class of subword complexes containing all cluster complexes of finite types. For the latter, the brick polytopes turn out to coincide with the known realizations of generalized associahedra, thus opening new perspectives on these constructions. This new approach yields in particular the vertex description of generalized associahedra, and a Minkowski
The diameter of associahedra
, 2014
"... It is proven here that the diameter of the ddimensional associahedron is 2d−4 when d is greater than 9. Two maximally distant vertices of this polytope are explicitly described as triangulations of a convex polygon, and their distance is obtained using combinatorial arguments. This settles two prob ..."
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Cited by 9 (2 self)
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It is proven here that the diameter of the ddimensional associahedron is 2d−4 when d is greater than 9. Two maximally distant vertices of this polytope are explicitly described as triangulations of a convex polygon, and their distance is obtained using combinatorial arguments. This settles two problems posed about twentyfive years ago by Daniel Sleator, Robert Tarjan, and William Thurston.
Geometric combinatorial algebras: cyclohedron and simplex
"... In this paper we report on results of our investigation into the algebraic structure supported by the combinatorial geometry of the cyclohedron. Our new graded algebra structures lie between two well known Hopf algebras: the MalvenutoReutenauer algebra of permutations and the LodayRonco algebra ..."
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In this paper we report on results of our investigation into the algebraic structure supported by the combinatorial geometry of the cyclohedron. Our new graded algebra structures lie between two well known Hopf algebras: the MalvenutoReutenauer algebra of permutations and the LodayRonco algebra of binary trees. Connecting algebra maps arise from a new generalization of the Tonks projection from the permutohedron to the associahedron, which we discover via the viewpoint of the graph associahedra of Carr and Devadoss. At the same time, that viewpoint allows exciting geometrical insights into the multiplicative structure of the algebras involved. Extending the Tonks projection also reveals a new graded algebra structure on the simplices. Finally this latter is extended to a new graded Hopf algebra with basis all the faces of the simplices.
Minkowski decompositions of associahedra
, 2011
"... Realisations of associahedra can be obtained from the classical permutahedron by removing some of its facets and the set of facets is determined by the diagonals of certain labeled convex planar ngons as shown by Hohlweg and Lange (2007). Ardila, Benedetti, and Doker (2010) expressed polytopes of ..."
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Realisations of associahedra can be obtained from the classical permutahedron by removing some of its facets and the set of facets is determined by the diagonals of certain labeled convex planar ngons as shown by Hohlweg and Lange (2007). Ardila, Benedetti, and Doker (2010) expressed polytopes of this type as Minkowski sums and differences of scaled faces of a standard simplex and computed the corresponding coefficients yI by Möbius inversion from the zI if tight righthand sides zI for all inequalities of the permutahedron are assumed. Given an associahedron of Hohlweg and Lange, we first characterise all tight values zI in terms of noncrossing diagonals of the associated labeled ngon, simplify the formula of Ardila et al., and characterise the remaining terms combinatorially.
Using spines to revisit a construction of the associahedron
, 2013
"... An associahedron is a polytope whose vertices correspond to triangulations of a convex polygon and whose edges correspond to flips between them. C. Hohlweg and C. Lange constructed various realizations of the associahedron, with relevant combinatorial properties in connection to the symmetric grou ..."
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An associahedron is a polytope whose vertices correspond to triangulations of a convex polygon and whose edges correspond to flips between them. C. Hohlweg and C. Lange constructed various realizations of the associahedron, with relevant combinatorial properties in connection to the symmetric group and to the classical permutahedron. We revisit this construction focussing on the spines of the triangulations, i.e. on their (oriented and labeled) dual trees. This new perspective leads to a noteworthy proof that these polytopes indeed realize the associahedron, and to new insights on various combinatorial properties of these realizations.
TypeB generalized triangulations and determinantal ideals
 Discrete Math
"... Abstract. For n ≥ 3, let Ωn be the set of line segments between the vertices of a convex ngon. For j ≥ 2, a jcrossing is a set of j line segments pairwise intersecting in the relative interior of the ngon. For k ≥ 1, let ∆ n,k be the simplicial complex of (typeA) generalized triangulations, i.e ..."
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Abstract. For n ≥ 3, let Ωn be the set of line segments between the vertices of a convex ngon. For j ≥ 2, a jcrossing is a set of j line segments pairwise intersecting in the relative interior of the ngon. For k ≥ 1, let ∆ n,k be the simplicial complex of (typeA) generalized triangulations, i.e. the simplicial complex of subsets of Ωn not containing any (k + 1)crossing. The complex ∆ n,k has been the central object of numerous papers. Here we continue this work by considering the complex of typeB generalized triangulations. For this we identify linesegments in Ω 2n which can be transformed into each other by a 180 • rotation of the 2ngon. Let Fn be the set Ω 2n after identification, then the complex D n,k of typeB generalized triangulations is the simplicial complex of subsets of Fn not containing any (k + 1)crossing in the above sense. For k = 1, we have that D n,1 is the simplicial complex of typeB triangulations of the 2ngon as defined in On the algebraical side we give a termorder on the monomials in the variables X ij , 1 ≤ i, j ≤ n, such that the corresponding initial ideal of the determinantal ideal generated by the (k + 1) times (k + 1) minors of the generic n × n matrix contains the StanleyReisner ideal of D n,k . We show that the minors form a GröbnerBasis whenever k ∈ {1, n − 2, n − 1} thereby proving the equality of both ideals and the unimodality of the hvector of the determinantal ideal in these cases. We conjecture this result to be true for all values of k < n.