Abstract. Let I ⊆ R = k[x 1 , ..., xn] be a homogeneous equigenerated ideal of degree r. We show here that the shapes of the Betti tables of the ideals I d stabilize, in the sense that there exists some D such that for all d ≥ D, β i,j+rd (I d ) = 0 ⇔ β i,j+rD (I D ) = 0. We also produce upper bounds for the stabilization index Stab(I). This strengthens the result of Cutkosky, Herzog, and Trung that the Castelnuovo-Mumford regularity reg(I d ) is eventually a linear function in d.

Given an edge ideal of graph G, we show that if the first nonlinear strand in the resolution of I G is zero until homological stage a 1 , then the next nonlinear strand in the resolution is zero until homological stage 2a 1 . Additionally, we define a sequence, called a jump sequence, characterizing the highest degrees of the free resolution of the edge ideal of G via the lower edge of the Betti diagrams of I G . These sequences strongly characterize topological properties of the underlying Stanley-Reisner complexes of edge ideals, and provide general conditions on construction of clique complexes on a fix set of vertices. We also provide an algorithm for obtaining a large class of realizable jump sequences and classes of Gorenstein edge ideals achieving high regularity.

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We introduce the package Posets for Macaulay2. This package provides a data structure and the necessary methods for working with partially ordered sets, also called posets. In particular, the package implements methods to enumerate many commonly studied classes of posets, perform operations on posets, and calculate various invariants associated to posets. INTRODUCTION.A partial order is a binary relation over a set P that is antisymmetric, reflexive, and transitive. A set P together with a partial order is called a poset, or partially ordered set. We refer the reader to the seminal text [Stanley 2012] for definitions omitted herein.Posets are combinatorial structures that are used in modern mathematical research, particularly in algebra. We introduce the package Posets for Macaulay2 [Grayson and Stillman] via three distinct posets or related ideals which arise naturally in combinatorial algebra.We first describe two posets that are generated from algebraic objects. The intersection semilattice associated to a hyperplane arrangement can be used to compute the number of unbounded and bounded real regions cut out by a hyperplane arrangement, as well as the dimensions of the homologies of the complex complement of a hyperplane arrangement.Given a monomial ideal, the lcm-lattice of its minimal generators gives information on the structure of the free resolution of the original ideal. Specifically, two monomial ideals with isomorphic lcm-lattices have the "same" (up to relabeling) minimal free resolution, and the lcm-lattice can be used to compute, among other things, the multigraded Betti numbers

Let M ⊆ k[x, y] be a monomial ideal M = (m 1 , m 2 , ..., m r ), where the m i are a minimal generating set of M . We construct an explicit free resolution of k over S = k[x, y]/M for all monomial ideals M , and provide recursive formulas for the Betti numbers. In particular, if M is any monomial ideal (excepting five degenerate cases), the total Betti numbers β S i (k) are given by β S 0 (k) = 1, β S 1 (k) = 2, and, where r is the number of minimal generators of M .This specializes to the classic example S = k[x, y]/(x 2 , xy), which has β S i (k) = f i+1 , where f i+1 is the (i+1)st Fibonacci number.