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183
Resolutions by mapping cones
 Homology Homotopy Appl
"... Many wellknown free resolutions arise as iterated mapping cones. Prominent examples are the EliahouKervaire resolution of stable monomial ideals (as noted by Evans and Charalambous [10]), and the Taylor resolution. The idea of the iterated mapping cone construction is the following: Let I ⊂ R be a ..."
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Cited by 33 (4 self)
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Many wellknown free resolutions arise as iterated mapping cones. Prominent examples are the EliahouKervaire resolution of stable monomial ideals (as noted by Evans and Charalambous [10]), and the Taylor resolution. The idea of the iterated mapping cone construction is the following: Let I ⊂ R be an ideal generated by
Lattice congruences, fans and Hopf algebras
 J. COMBIN. THEORY SER. A
, 2004
"... 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 ..."
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Cited by 31 (12 self)
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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.
Realizations of the associahedron and cyclohedron
, 2005
"... We describe many different realizations with integer coordinates for the associahedron (i.e. the Stasheff polytope) and for the cyclohedron (i.e. the BottTaubes polytope) and compare them to the permutahedron of type A and B respectively. The coordinates are obtained by an algorithm which uses an o ..."
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Cited by 31 (5 self)
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We describe many different realizations with integer coordinates for the associahedron (i.e. the Stasheff polytope) and for the cyclohedron (i.e. the BottTaubes polytope) and compare them to the permutahedron of type A and B respectively. The coordinates are obtained by an algorithm which uses an oriented Coxeter graph of type An or Bn respectively as only input and which specialises to a procedure presented by J.L. Loday for a certain orientation of An.
Shifted simplicial complexes are Laplacian integral
 Trans. Amer. Math. Soc
, 2002
"... Abstract. We show that the combinatorial Laplace operators associated to the boundary maps in a shifted simplicial complex have all integer spectra. We give a simple combinatorial interpretation for the spectra in terms of vertex degree sequences, generalizing a theorem of Merris for graphs. We also ..."
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Cited by 30 (7 self)
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Abstract. We show that the combinatorial Laplace operators associated to the boundary maps in a shifted simplicial complex have all integer spectra. We give a simple combinatorial interpretation for the spectra in terms of vertex degree sequences, generalizing a theorem of Merris for graphs. We also conjecture a majorization inequality for the spectra of these Laplace operators in an arbitrary simplicial complex, with equality achieved if and only if the complex is shifted. This generalizes a conjecture of Grone and Merris for graphs. 1.
Algebraic Shifting and Sequentially CohenMacaulay Simplicial Complexes
 ELECTRONIC JOURNAL OF COMBINATORICS
, 1996
"... Björner and Wachs generalized the definition of shellability by dropping the assumption of purity; they also introduced the htriangle, a doublyindexed generalization of the hvector which is combinatorially significant for nonpure shellable complexes. Stanley subsequently defined a nonpure simp ..."
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Cited by 29 (0 self)
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Björner and Wachs generalized the definition of shellability by dropping the assumption of purity; they also introduced the htriangle, a doublyindexed generalization of the hvector which is combinatorially significant for nonpure shellable complexes. Stanley subsequently defined a nonpure simplicial complex to be sequentially CohenMacaulay if it satisfies algebraic conditions that generalize the CohenMacaulay conditions for pure complexes, so that a nonpure shellable complex is sequentially CohenMacaulay. We show that algebraic shifting preserves the htriangle of a simplicial complex K if and only if K is sequentially CohenMacaulay. This generalizes a result of Kalai's for the pure case. Immediate consequences include that nonpure shellable complexes and sequentially CohenMacaulay complexes have the same set of possible htriangles.
Vertex decomposable graphs and obstructions to shellability
, 2009
"... Inspired by several recent papers on the edge ideal of a graph G, we study the equivalent notion of the independence complex of G. Using the tool of vertex decomposability from geometric combinatorics, we show that 5chordal graphs with no chordless 4cycles are shellable and sequentially CohenMac ..."
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Cited by 29 (2 self)
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Inspired by several recent papers on the edge ideal of a graph G, we study the equivalent notion of the independence complex of G. Using the tool of vertex decomposability from geometric combinatorics, we show that 5chordal graphs with no chordless 4cycles are shellable and sequentially CohenMacaulay. We use this result to characterize the obstructions to shellability in flag complexes, extending work of Billera, Myers, and Wachs. We also show how vertex decomposability may be used to show that certain graph constructions preserve shellability.
Characteristic polynomials of subspace arrangements and finite fields
 Advances in Math. 122
, 1996
"... Let A be any subspace arrangement in Rn defined over the integers and let Fq denote the finite field with q elements. Let q be a large prime. We prove that the characteristic polynomial /(A, q) of A counts the number of points in Fnq that do not lie in any of the subspaces of A, viewed as subsets of ..."
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Cited by 28 (6 self)
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Let A be any subspace arrangement in Rn defined over the integers and let Fq denote the finite field with q elements. Let q be a large prime. We prove that the characteristic polynomial /(A, q) of A counts the number of points in Fnq that do not lie in any of the subspaces of A, viewed as subsets of Fnq. This observation, which generalizes a theorem of Blass and Sagan about subarrangements of the Bn arrangement, reduces the computation of /(A, q) to a counting problem and provides an explanation for the wealth of combinatorial results discovered in the theory of hyperplane arrangements in recent years. The basic idea has its origins in the work of Crapo and Rota (1970). We find new classes of hyperplane arrangements whose characteristic polynomials have simple form and very often factor completely over the nonnegative integers. 1996 Academic Press, Inc. 1.
Cambrian lattices
, 2004
"... For an arbitrary finite Coxeter group W, we define the family of Cambrian lattices for W as quotients of the weak order on W with respect to certain lattice congruences. We associate to each Cambrian lattice a complete fan, which we conjecture is the normal fan of a polytope combinatorially isomorp ..."
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Cited by 25 (5 self)
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For an arbitrary finite Coxeter group W, we define the family of Cambrian lattices for W as quotients of the weak order on W with respect to certain lattice congruences. We associate to each Cambrian lattice a complete fan, which we conjecture is the normal fan of a polytope combinatorially isomorphic to the generalized associahedron for W. In types A and B, we obtain, by means of a fiberpolytope construction, combinatorial realizations of the Cambrian lattices in terms of triangulations and in terms of permutations. Using this combinatorial information, we prove in types A and B that the Cambrian fans are combinatorially isomorphic to the normal fans of the generalized associahedra and that one of the Cambrian fans is linearly isomorphic to Fomin and Zelevinsky’s construction of the normal fan as a “cluster fan.” Our construction does not require a crystallographic Coxeter group and therefore suggests a definition, at least on the level of cellular spheres, of a generalized associahedron for any finite Coxeter group. The Tamari lattice is one of the Cambrian lattices of type A, and two “Tamari ” lattices in type B are identified and characterized in terms of signed pattern avoidance. We also show that open intervals in Cambrian lattices are either contractible or homotopy equivalent to spheres.
How To Shell A Monoid
 MATH. ANN
, 1997
"... We study monoid algebras which possess an initial ideal that is the StanleyReisner ideal of a poset. We construct a quadratic noncommutative Gröbner basis which induces nonpure shellings for all finite intervals of the monoid. These shellings express the Poincare series of the infinite resolut ..."
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Cited by 24 (3 self)
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We study monoid algebras which possess an initial ideal that is the StanleyReisner ideal of a poset. We construct a quadratic noncommutative Gröbner basis which induces nonpure shellings for all finite intervals of the monoid. These shellings express the Poincare series of the infinite resolution of the residue class field as a rational function. Our results apply to the coordinate rings of many toric varieties, including all affine normal toric surfaces. In the latter case we construct an explicit minimal free resolution of the residue field.