Associated to any simplicial complex \Delta on n vertices is a square-free monomial ideal I \Delta in the polynomial ring A = k[x 1 ; : : : ; xn ], and its quotient k[\Delta] = A=I \Delta known as the Stanley-Reisner 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 A-module if and only if \Delta is Cohen-Macaulay over k, and show how to compute the Betti numbers dim k Tor i (k[\Delta]; k) in some cases where \Delta is well-behaved (shellable, Cohen-Macaulay, or Buchsbaum). Some other applications of the notion of shellability are also discussed.

Abstract not found

Abstract. We consider a specialization YM(q, t) of the Tutte polynomial of a matroid M which is inspired by analogy with the Potts model from statistical mechanics. The only information lost in this specialization is the number of loops of M. We show that the coefficients of YM(1 − p, t) are very simply related to the ranks of the Whitney homology groups of the opposite partial orders of the independent set complexes of the duals of the truncations of M. In particular, we obtain a new homological interpretation for the coefficients of the characteristic polynomial of a matroid. 0. Introduction. In 1954, Tutte [30] introduced the dichromate of a (finite) graph, which has since become known as the Tutte polynomial. In the four decades since then this has provided a profound link between combinatorics and other branches of mathematics

One of the most important numerical quantities that can be computed from a graph G is the two-variable 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 depth-first search.

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 Orlik-Solomon 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 E-module N called the singular variety,

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 M-shellability, which is a generalization to pure posets of the property of shellability of simplicial complexes, and derive inequalities that the rank-numbers of M-shellable posets must satisfy. We also introduce a decomposition property for simplicial complexes called a convex ear-decomposition, and, using results of Kalai and Stanley on h-vectors of simplicial polytopes, we show that h-vectors of pure rank-d 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 ear-decomposition. This enables us to construct an associated M-shellable poset, whose set of rank-numbers is the h-vector of the matroid complex. This results in a combinatorial proof of a conjecture of Hibi [17] that the h-vector of a matroid complex satisfies the above two sets of inequalities. 1.

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 Stanley-Reisner ideals I such that is the combinatorial Alexander-dual of a non-pure shellable simplicial complex. For generic monomial ideals the given resolution is the resolution determined by the Scarf-complex as shown by Bayer & Sturmfels and Bayer, Peeva & Sturmfels and by Miller, Sturmfels & Yanagawa in its most general form. For stable and squarefree-stable 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 non-pure shellable. This is a condition which implies in the pure case, in other words k[] is homogeneous, that k[] is Koszul. 1.

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-set 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 independent-set 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 alternating-sign property for the Mayer coefficients. We conclude with a brief discussion of the repulsive lattice gas on countably infinite graphs.

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.

Constructibility is a condition on pure simplicial complexes that is weaker than shellability. In this paper we show that non-constructible triangulations of the d-dimensional sphere exist for every d 3. This answers a question of Danaraj & Klee [8]; it also strengthens a result of Lickorish [13] about non-shellable spheres. Furthermore, we provide a hierarchy of combinatorial decomposition properties that follow from the existence of a non-trivial knot with "few edges" in a 3-sphere or 3-ball, and a similar hierarchy for 3-balls with a knotted spanning arc that consists of "few edges."