Results 21  30
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183
Poset fiber theorems
 TRANS. AMER. MATH. SOC
, 2004
"... Suppose that f: P → Q is a poset map whose fibers f −1 (Q≤q) are sufficiently well connected. Our main result is a formula expressing the homotopy type of P in terms of Q and the fibers. Several fiber theorems from the literature (due to Babson, Baclawski and Quillen) are obtained as consequences o ..."
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Cited by 23 (3 self)
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Suppose that f: P → Q is a poset map whose fibers f −1 (Q≤q) are sufficiently well connected. Our main result is a formula expressing the homotopy type of P in terms of Q and the fibers. Several fiber theorems from the literature (due to Babson, Baclawski and Quillen) are obtained as consequences or special cases. Homology, CohenMacaulay, and equivariant versions are given, and some applications are discussed.
Chordal and sequentially cohenmacaulay clutters
"... We extend the definition of chordal from graphs to clutters. The resulting family generalizes both chordal graphs and matroids, and obeys many of the same algebraic and geometric properties. Specifically, the independence complex of a chordal clutter is shellable, hence sequentially CohenMacaulay; ..."
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Cited by 21 (1 self)
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We extend the definition of chordal from graphs to clutters. The resulting family generalizes both chordal graphs and matroids, and obeys many of the same algebraic and geometric properties. Specifically, the independence complex of a chordal clutter is shellable, hence sequentially CohenMacaulay; and the circuit ideal of a certain complement to such a clutter has a linear resolution. Minimal nonchordal clutters are also closely related to obstructions to shellability, and we give some general families of such obstructions, together with a classification by computation of all obstructions to shellability on 6 vertices. 1
The homology representations of the kequal partition lattice
 Trans. Amer. Math. Soc
, 1994
"... Abstract. We determine the character of the action of the symmetric group on the homology of the induced subposet of the lattice of partitions of the set {1, 2,...,n}obtained by restricting block sizes to the set {1,k,k+1,...}.A plethystic formula for the generating function of the Frobenius charact ..."
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Cited by 21 (9 self)
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Abstract. We determine the character of the action of the symmetric group on the homology of the induced subposet of the lattice of partitions of the set {1, 2,...,n}obtained by restricting block sizes to the set {1,k,k+1,...}.A plethystic formula for the generating function of the Frobenius characteristic of the representation is given. We combine techniques from the theory of nonpure shellability, recently developed by Björner and Wachs, with symmetric function techniques, developed by Sundaram, for determining representations on the homology of subposets of the partition lattice. For 2 ≤ k ≤ n, the kequal partition lattice Π (k,1 n−k) is defined to be the joinsublattice of the partition lattice Πn generated by set partitions consisting of n−k blocks of size one and one block of size k. In other words, Π (k,1 n−k) is the joinsublattice of Πn consisting of partitions having no blocks of sizes 2, 3,...,k−1. This
Structure of the LodayRonco Hopf algebra of trees
 J. ALGEBRA
"... Loday and Ronco defined an interesting Hopf algebra structure on the linear span of the set of planar binary trees. They showed that the inclusion of the Hopf algebra of noncommutative symmetric functions in the MalvenutoReutenauer Hopf algebra of permutations factors through their Hopf algebra ..."
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Cited by 21 (3 self)
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Loday and Ronco defined an interesting Hopf algebra structure on the linear span of the set of planar binary trees. They showed that the inclusion of the Hopf algebra of noncommutative symmetric functions in the MalvenutoReutenauer Hopf algebra of permutations factors through their Hopf algebra of trees, and these maps correspond to natural maps from the weak order on the symmetric group to the Tamari order on planar binary trees to the boolean algebra. We further study the structure of this Hopf algebra of trees using a new basis for it. We describe the product, coproduct, and antipode in terms of this basis and use these results to elucidate its Hopfalgebraic structure. We also obtain a transparent proof of its isomorphism with the noncommutative ConnesKreimer Hopf algebra of Foissy, and show that this algebra is related to noncommutative symmetric functions as the (commutative) ConnesKreimer Hopf algebra is related to symmetric functions.
Complexes of not iconnected graphs
 TOPOLOGY
, 1999
"... Complexes of (not) connected graphs, hypergraphs and their homology appear in the construction of knot invariants given by V. Vassiliev [V1, V2, V3]. In this paper we study the complexes of not iconnected khypergraphs on n vertices. We show that the complex of not 2connected graphs has the homot ..."
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Complexes of (not) connected graphs, hypergraphs and their homology appear in the construction of knot invariants given by V. Vassiliev [V1, V2, V3]. In this paper we study the complexes of not iconnected khypergraphs on n vertices. We show that the complex of not 2connected graphs has the homotopy type of a wedge of (n − 2)! spheres of dimension 2n − 5. This answers one of the questions raised by Vassiliev [V3] in connection with knot invariants. For this case the Snaction on the homology of the complex is also determined. For complexes of not 2connected khypergraphs we provide a formula for the generating function of the Euler characteristic, and we introduce certain lattices of graphs that encode their topology. We also present partial results for some other cases. In particular, we show that the complex of not (n − 2)connected graphs is Alexander dual to the complex of partial matchings of the complete graph. For not (n − 3)connected graphs we provide a formula for the generating function of the Euler characteristic.
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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Cited by 21 (5 self)
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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
Lattice Congruences of the Weak Order
 ORDER
, 2004
"... We study the congruence lattice of the poset of regions of a hyperplane arrangement, with particular emphasis on the weak order on a finite Coxeter group. Our starting point is a theorem from a previous paper which gives a geometric description of the poset of joinirreducibles of the congruence la ..."
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Cited by 20 (9 self)
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We study the congruence lattice of the poset of regions of a hyperplane arrangement, with particular emphasis on the weak order on a finite Coxeter group. Our starting point is a theorem from a previous paper which gives a geometric description of the poset of joinirreducibles of the congruence lattice of the poset of regions in terms of certain polyhedral decompositions of the hyperplanes. For a finite Coxeter system (W,S) and a subset K ⊆ S, letηK: w ↦ → wK be the projection onto the parabolic subgroup WK. We show that the fibers of ηK constitute the smallest lattice congruence with 1 ≡ s for every s ∈ (S − K). We give an algorithm for determining the congruence lattice of the weak order for any finite Coxeter group and for a finite Coxeter group of type A or B we define a directed graph on subsets or signed subsets such that the transitive closure of the directed graph is the poset of joinirreducibles of the congruence lattice of the weak order.
Why the characteristic polynomial factors
 BULL. AMER. MATH. SOC
, 1999
"... We survey three methods for proving that the characteristic polynomial of a finite ranked lattice factors over the nonnegative integers and indicate how they have evolved recently. The first technique uses geometric ideas and is based on Zaslavsky’s theory of signed graphs. The second approach is a ..."
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Cited by 20 (2 self)
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We survey three methods for proving that the characteristic polynomial of a finite ranked lattice factors over the nonnegative integers and indicate how they have evolved recently. The first technique uses geometric ideas and is based on Zaslavsky’s theory of signed graphs. The second approach is algebraic and employs results of Saito and Terao about free hyperplane arrangements. Finally we consider a purely combinatorial theorem of Stanley about supersolvable lattices and its generalizations.
General Lexicographic Shellability And Orbit Arrangements
 Ann. of Comb
, 1996
"... We introduce a new poset property which we call ECshellability. It is more general than the more established concept of ELshellability, but still implies shellability. Because of the Theorem 3.10 ECshellability is entitled to be called general lexicographic shellability. As an application of ..."
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Cited by 18 (4 self)
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We introduce a new poset property which we call ECshellability. It is more general than the more established concept of ELshellability, but still implies shellability. Because of the Theorem 3.10 ECshellability is entitled to be called general lexicographic shellability. As an application of our new concept, we prove that intersection lattices \Pi of orbit arrangements A are ECshellable for a very large class of partitions . This allows to compute the topology of the link and the complement for these arrangements. In particular, for this class of 's, we are able to settle a conjecture of A.Bjorner, stating that the cohomologygroups of the complement of the orbit arrangements are torsionfree. We also present a class of partitions for which \Pi is not shellable, along with a few smaller things scattered throughout the paper. 1.