Results 1  10
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81
Resolutions Of StanleyReisner Rings And Alexander Duality
, 1996
"... Associated to any simplicial complex \Delta on n vertices is a squarefree monomial ideal I \Delta in the polynomial ring A = k[x 1 ; : : : ; xn ], and its quotient k[\Delta] = A=I \Delta known as the StanleyReisner ring. This note considers a simplicial complex which is in a sense a canonical ..."
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Cited by 78 (2 self)
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Associated to any simplicial complex \Delta on n vertices is a squarefree monomial ideal I \Delta in the polynomial ring A = k[x 1 ; : : : ; xn ], and its quotient k[\Delta] = A=I \Delta known as the StanleyReisner ring. This note considers a simplicial complex which is in a sense a canonical Alexander dual to \Delta, previously considered in [Ba, BrHe]. Using Alexander duality and a result of Hochster computing the Betti numbers dim k Tor i (k[\Delta]; k), it is shown (Proposition 1) that these Betti numbers are computable from the homology of links of faces in \Delta . As corollaries, we prove that I \Delta has a linear resolution as Amodule if and only if \Delta is CohenMacaulay over k, and show how to compute the Betti numbers dim k Tor i (k[\Delta]; k) in some cases where \Delta is wellbehaved (shellable, CohenMacaulay, or Buchsbaum). Some other applications of the notion of shellability are also discussed.
Bounds On The Complex Zeros Of (Di)Chromatic Polynomials And PottsModel Partition Functions
 Chromatic Roots Are Dense In The Whole Complex Plane, Combinatorics, Probability and Computing
"... I show that there exist universal constants C(r) < ∞ such that, for all loopless graphs G of maximum degree ≤ r, the zeros (real or complex) of the chromatic polynomial PG(q) lie in the disc q  < C(r). Furthermore, C(r) ≤ 7.963907r. This result is a corollary of a more general result on the zeros ..."
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Cited by 47 (11 self)
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I show that there exist universal constants C(r) < ∞ such that, for all loopless graphs G of maximum degree ≤ r, the zeros (real or complex) of the chromatic polynomial PG(q) lie in the disc q  < C(r). Furthermore, C(r) ≤ 7.963907r. This result is a corollary of a more general result on the zeros of the Pottsmodel partition function ZG(q, {ve}) in the complex antiferromagnetic regime 1 + ve  ≤ 1. The proof is based on a transformation of the Whitney–Tutte–Fortuin–Kasteleyn representation of ZG(q, {ve}) to a polymer gas, followed by verification of the Dobrushin–Koteck´y–Preiss condition for nonvanishing of a polymermodel partition function. I also show that, for all loopless graphs G of secondlargest degree ≤ r, the zeros of PG(q) lie in the disc q  < C(r) + 1. KEY WORDS: Graph, maximum degree, secondlargest degree, chromatic polynomial,
The Bergman complex of a matroid and phylogenetic trees
 the Journal of Combinatorial Theory, Series B. arXiv:math.CO/0311370
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Lattice path matroids: enumerative aspects and Tutte polynomials
 J. Combin. Theory Ser. A
, 2003
"... Abstract. Fix two lattice paths P and Q from (0, 0) to (m, r) that use East and North steps with P never going above Q. We show that the lattice paths that go from (0, 0) to (m, r) and that remain in the region bounded by P and Q can be identified with the bases of a particular type of transversal m ..."
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Cited by 24 (9 self)
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Abstract. Fix two lattice paths P and Q from (0, 0) to (m, r) that use East and North steps with P never going above Q. We show that the lattice paths that go from (0, 0) to (m, r) and that remain in the region bounded by P and Q can be identified with the bases of a particular type of transversal matroid, which we call a lattice path matroid. We consider a variety of enumerative aspects of these matroids and we study three important matroid invariants, namely the Tutte polynomial and, for special types of lattice path matroids, the characteristic polynomial and the β invariant. In particular, we show that the Tutte polynomial is the generating function for two basic lattice path statistics and we show that certain sequences of lattice path matroids give rise to sequences of Tutte polynomials for which there are relatively simple generating functions. We show that Tutte polynomials of lattice path matroids can be computed in polynomial time. Also, we obtain a new result about lattice paths from an analysis of the β invariant of certain lattice path matroids. 1.
Hyperplane arrangement cohomology and monomials in the exterior algebra
, 2000
"... Abstract. We show that if X is the complement of a complex hyperplane arrangement, then the homology of X has linear free resolution as a module over the exterior algebra on the first cohomology of X. We study invariants of X that can be deduced from this resolution. A key ingredient is a result of ..."
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Cited by 22 (4 self)
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Abstract. We show that if X is the complement of a complex hyperplane arrangement, then the homology of X has linear free resolution as a module over the exterior algebra on the first cohomology of X. We study invariants of X that can be deduced from this resolution. A key ingredient is a result of Aramova, Avramov, and Herzog (2000) on resolutions of monomial ideals in the exterior algebra. We give a new conceptual proof of this result. Let X be the complement of a complex hyperplane arrangement A. Inthispaper we study the singular homology H∗(X) as a module over the exterior algebra E on the first singular cohomology V: = H1 (X) always with coefficients in a fixed field K. Our first main result (Section 1) asserts that H∗(X) is generated in a single degree and has a linear free resolution; this amounts to an infinite sequence of statements about the multiplication in the OrlikSolomon algebra H ∗ (X). We also analyze other topological examples from the point of view of resolutions over the exterior algebra. In Section 2 we study an invariant of an Emodule N called the singular variety,
The Tutte polynomial of a graph, depthfirst search, and simplicial complex partitions
 ELECTRONIC J. COMBINATORICS
, 1996
"... One of the most important numerical quantities that can be computed from a graph G is the twovariable Tutte polynomial. Specializations of the Tutte polynomial count various objects associated with G, e.g., subgraphs, spanning trees, acyclic orientations, inversions and parking functions. We show t ..."
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Cited by 21 (3 self)
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One of the most important numerical quantities that can be computed from a graph G is the twovariable Tutte polynomial. Specializations of the Tutte polynomial count various objects associated with G, e.g., subgraphs, spanning trees, acyclic orientations, inversions and parking functions. We show that by partitioning certain simplicial complexes related to G into intervals, one can provide combinatorial demonstrations of these results. One of the primary tools for providing such a partition is depthfirst search.
The repulsive lattice gas, the independentset polynomial, and the Lovász local lemma
, 2004
"... We elucidate the close connection between the repulsive lattice gas in equilibrium statistical mechanics and the Lovász local lemma in probabilistic combinatorics. We show that the conclusion of the Lovász local lemma holds for dependency graph G and probabilities {px} if and only if the independent ..."
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Cited by 21 (7 self)
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We elucidate the close connection between the repulsive lattice gas in equilibrium statistical mechanics and the Lovász local lemma in probabilistic combinatorics. We show that the conclusion of the Lovász local lemma holds for dependency graph G and probabilities {px} if and only if the independentset polynomial for G is nonvanishing in the polydisc of radii {px}. Furthermore, we show that the usual proof of the Lovász local lemma — which provides a sufficient condition for this to occur — corresponds to a simple inductive argument for the nonvanishing of the independentset polynomial in a polydisc, which was discovered implicitly by Shearer [98] and explicitly by Dobrushin [37, 38]. We also present some refinements and extensions of both arguments, including a generalization of the Lovász local lemma that allows for “soft” dependencies. In addition, we prove some general properties of the partition function of a repulsive lattice gas, most of which are consequences of the alternatingsign property for the Mayer coefficients. We conclude with a brief discussion of the repulsive lattice gas on countably infinite graphs.
MATROID POLYTOPES, NESTED SETS AND BERGMAN FANS
, 2004
"... The tropical variety defined by linear equations with constant coefficients is the Bergman fan of the corresponding matroid. Building on a selfcontained introduction to matroid polytopes, we present a geometric construction of the Bergman fan, and we discuss its relationship with the simplicial com ..."
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Cited by 21 (3 self)
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The tropical variety defined by linear equations with constant coefficients is the Bergman fan of the corresponding matroid. Building on a selfcontained introduction to matroid polytopes, we present a geometric construction of the Bergman fan, and we discuss its relationship with the simplicial complex of nested sets in the lattice of flats. The Bergman complex is triangulated by the nested set complex, and the two complexes coincide if and only if every connected flat remains connected after contracting along any subflat. This sharpens a result of ArdilaKlivans who showed that the Bergman complex is triangulated by the order complex of the lattice of flats. The nested sets specify the De ConciniProcesi compactification of the complement of a hyperplane arrangement, while the Bergman fan specifies the tropical compactification. These two compactifications are almost equal, and we highlight the subtle differences.
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.
Discrete Morse Theory for Cellular Resolutions
 J. Reine Angew. Math
, 2000
"... We develop an analog of Forman's discrete Morse theory for cell complexes in the setting of cellular resolutions of multigraded monomial modules. In particular, using discrete Morse theory for cellular resolutions of multigraded ideals we are able to give minimal cellular resolutions for generic ..."
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Cited by 17 (2 self)
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We develop an analog of Forman's discrete Morse theory for cell complexes in the setting of cellular resolutions of multigraded monomial modules. In particular, using discrete Morse theory for cellular resolutions of multigraded ideals we are able to give minimal cellular resolutions for generic and shellable monomial modules. The latter ones are introduced in this paper and include stable, squarefree stable monomial ideals and StanleyReisner ideals I such that is the combinatorial Alexanderdual of a nonpure shellable simplicial complex. For generic monomial ideals the given resolution is the resolution determined by the Scarfcomplex as shown by Bayer & Sturmfels and Bayer, Peeva & Sturmfels and by Miller, Sturmfels & Yanagawa in its most general form. For stable and squarefreestable monomial ideals the minimal resolution had been determined by Eliahou & Kervaire respectively Aramova, Herzog & Hibi and Peeva by algebraic means. In great detail we examplify our method for powers of the maximal ideal. Parallel to our work Skoldberg has constructed minimal free resolutions for a class of ideals that includes the shellable ones by algebraic means. The preceding constructions can be viewed as a generalization of a result by Lyubeznik which gives a resolution that is a subcomplex of the Taylor resolution. For ane semigroup rings k[] we are able to give a minimal free resolution of the eld k in case all lower intervals in the poset are nonpure shellable. This is a condition which implies in the pure case, in other words k[] is homogeneous, that k[] is Koszul. 1.