Results 1  10
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59
Shellable and CohenMacaulay partially ordered sets
 Trans. Amer. Math. Soc
, 1980
"... Abstract. In this paper we study shellable posets (partially ordered sets), that is, finite posets such that the simplicial complex of chains is shellable. It is shown that all admissible lattices (including all finite semimodular and supersolvable lattices) and all bounded locally semimodular finit ..."
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Cited by 122 (5 self)
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Abstract. In this paper we study shellable posets (partially ordered sets), that is, finite posets such that the simplicial complex of chains is shellable. It is shown that all admissible lattices (including all finite semimodular and supersolvable lattices) and all bounded locally semimodular finite posets are shellable. A technique for labeling the edges of the Hasse diagram of certain lattices, due to R. Stanley, is generalized to posets and shown to imply shellability, while Stanley's main theorem on the JordanHolder sequences of such labelings remains valid. Further, we show a number of ways in which shellable posets can be constructed from other shellable posets and complexes. These results give rise to several new examples of CohenMacaulay posets. For instance, the lattice of subgroups of a finite group G is CohenMacaulay (in fact shellable) if and only if G is supersolvable. Finally, it is shown that all the higher order complexes of a finite planar distributive lattice are shellable. Introduction. A pure finite simplicial complex A is said to be shellable if its maximal faces can be ordered F,, F2,..., Fn in such a way that Fk n ( U *j / Fj) is
On deletion in Delaunay triangulation
 Internat. J. Comput. Geom. Appl
, 2002
"... This paper presents how the space of spheres and shelling may be used to delete a point from a ddimensional triangulation efficiently. In dimension two, if k is the degree of the deleted vertex, the complexity is O(k log k), but we notice that this number only applies to low cost operations, while ..."
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Cited by 46 (4 self)
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This paper presents how the space of spheres and shelling may be used to delete a point from a ddimensional triangulation efficiently. In dimension two, if k is the degree of the deleted vertex, the complexity is O(k log k), but we notice that this number only applies to low cost operations, while time consuming computations are only done a linear number of times. This algorithm may be viewed as a variation of Heller’s algorithm,[1, 2] which is popular in the geographic information system community. Unfortunately, Heller algorithm is false, as explained in this paper.
Two decompositions in topological combinatorics with applications to matroid complexes
 Trans. Amer. Math. Soc
, 1997
"... Abstract. This paper introduces two new decomposition techniques which are related to the classical notion of shellability of simplicial complexes, and uses the existence of these decompositions to deduce certain numerical properties for an associated enumerative invariant. First, we introduce the n ..."
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Cited by 20 (1 self)
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Abstract. This paper introduces two new decomposition techniques which are related to the classical notion of shellability of simplicial complexes, and uses the existence of these decompositions to deduce certain numerical properties for an associated enumerative invariant. First, we introduce the notion of Mshellability, which is a generalization to pure posets of the property of shellability of simplicial complexes, and derive inequalities that the ranknumbers of Mshellable posets must satisfy. We also introduce a decomposition property for simplicial complexes called a convex eardecomposition, and, using results of Kalai and Stanley on hvectors of simplicial polytopes, we show that hvectors of pure rankd simplicial complexes that have this property satisfy h0 ≤ h1 ≤ ·· · ≤ h [d/2] and hi ≤ hd−i for 0 ≤ i ≤ [d/2]. We then show that the abstract simplicial complex formed by the collection of independent sets of a matroid (or matroid complex) admits a special type of convex eardecomposition called a PS eardecomposition. This enables us to construct an associated Mshellable poset, whose set of ranknumbers is the hvector of the matroid complex. This results in a combinatorial proof of a conjecture of Hibi [17] that the hvector of a matroid complex satisfies the above two sets of inequalities. 1.
Randomized Simplex Algorithms on KleeMinty Cubes
 COMBINATORICA
, 1994
"... We investigate the behavior of randomized simplex algorithms on special linear programs. For this, we use combinatorial models for the KleeMinty cubes [22] and similar linear programs with exponential decreasing paths. The analysis of two most natural randomized pivot rules on the KleeMinty cubes ..."
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Cited by 19 (6 self)
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We investigate the behavior of randomized simplex algorithms on special linear programs. For this, we use combinatorial models for the KleeMinty cubes [22] and similar linear programs with exponential decreasing paths. The analysis of two most natural randomized pivot rules on the KleeMinty cubes leads to (nearly) quadratic lower bounds for the complexity of linear programming with random pivots. Thus we disprove two bounds (for the expected running time of the randomedge simplex algorithm on KleeMinty cubes) conjectured in the literature. At the same time, we establish quadratic upper bounds for the expected length of a path for a simplex algorithm with random pivots on the classes of linear programs under investigation. In contrast to this, we find that the average length of an increasing path in a KleeMinty cube is exponential when all paths are taken with equal probability.
Lattice congruences, fans and Hopf algebras
 J. Combin. Theory Ser. A
"... Abstract. We give a unified explanation of the geometric and algebraic properties of two wellknown maps, one from permutations to triangulations, and another from permutations to subsets. Furthermore we give a broad generalization of the maps. Specifically, for any lattice congruence of the weak or ..."
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Cited by 17 (8 self)
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Abstract. We give a unified explanation of the geometric and algebraic properties of two wellknown maps, one from permutations to triangulations, and another from permutations to subsets. Furthermore we give a broad generalization of the maps. Specifically, for any lattice congruence of the weak order on a Coxeter group we construct a complete fan of convex cones with strong properties relative to the corresponding lattice quotient of the weak order. We show that if a family of lattice congruences on the symmetric groups satisfies certain compatibility conditions then the family defines a sub Hopf algebra of the MalvenutoReutenauer Hopf algebra of permutations. Such a sub Hopf algebra has a basis which is described by a type of patternavoidance. Applying these results, we build the MalvenutoReutenauer algebra as the limit of an infinite sequence of smaller algebras, where the second algebra in the sequence is the Hopf algebra of noncommutative symmetric functions. We also associate both a fan and a Hopf algebra to a set of permutations which appears to be equinumerous with the Baxter permutations. 1.
Decompositions of Simplicial Balls and Spheres With Knots Consisting of Few Edges
, 1999
"... Constructibility is a condition on pure simplicial complexes that is weaker than shellability. In this paper we show that nonconstructible triangulations of the ddimensional sphere exist for every d 3. This answers a question of Danaraj & Klee [8]; it also strengthens a result of Lickorish [13] a ..."
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Cited by 16 (6 self)
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Constructibility is a condition on pure simplicial complexes that is weaker than shellability. In this paper we show that nonconstructible triangulations of the ddimensional sphere exist for every d 3. This answers a question of Danaraj & Klee [8]; it also strengthens a result of Lickorish [13] about nonshellable spheres. Furthermore, we provide a hierarchy of combinatorial decomposition properties that follow from the existence of a nontrivial knot with "few edges" in a 3sphere or 3ball, and a similar hierarchy for 3balls with a knotted spanning arc that consists of "few edges."
Projectivities in simplicial complexes and colorings of simple polytopes
 Math. Z
"... For each strongly connected finitedimensional (pure) simplicial complex ∆ we construct a finite group Π(∆), the group of projectivities of ∆, which is a combinatorial but not a topological invariant of ∆. This group is studied for simplicial manifolds and, in particular, for polytopal simplicial sp ..."
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Cited by 15 (5 self)
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For each strongly connected finitedimensional (pure) simplicial complex ∆ we construct a finite group Π(∆), the group of projectivities of ∆, which is a combinatorial but not a topological invariant of ∆. This group is studied for simplicial manifolds and, in particular, for polytopal simplicial spheres. The results are used to solve a coloring problem for simplicial (or, dually, simple) polytopes which arose in the area of toric algebraic varieties. 1
A partial order on the regions of R n dissected by hyperplanes
 Trans. Amer. Math. Soc
, 1984
"... Abstract. We study a partial order on the regions of R " dissected by hyperplanes. This includes a computation of the Mobius function and, in some cases, of the homotopy type. Applications are presented to zonotopes, the weak Bruhat order on Weyl groups and acyclic orientations of graphs. 0. Introdu ..."
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Cited by 13 (0 self)
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Abstract. We study a partial order on the regions of R " dissected by hyperplanes. This includes a computation of the Mobius function and, in some cases, of the homotopy type. Applications are presented to zonotopes, the weak Bruhat order on Weyl groups and acyclic orientations of graphs. 0. Introduction. Let % — {77,, H2,...,Hk] be a set of hyperplanes in R". Then the components of R " — UHe.xH form a set 9t of open ncells we will call regions. Traditionally 9t has been studied in terms of enumeration, for instance counting the number of regions and the number of intersections of various dimensions among the hyperplanes in %. For a thorough discussion of this problem see Zaslavsky's