Caroline J. Klivans

Learn More
We study the Bergman complex B(M) of a matroid M : a polyhedral complex which arises in algebraic geometry, but which we describe purely combinatorially. We prove that a natural subdivision of the Bergman complex of M is a geometric realization of the order complex of its lattice of flats. In addition, we show that the Bergman fan B̃(Kn) of the graphical(More)
Abstract. We generalize the definition and enumeration of spanning trees from the setting of graphs to that of arbitrary-dimensional simplicial complexes ∆, extending an idea due to G. Kalai. We prove a simplicial version of the Matrix-Tree Theorem that counts simplicial spanning trees, weighted by the squares of the orders of their top-dimensional integral(More)
We study, in three parts, degree sequences of k-families (or k-uniform hypergraphs) and shifted k-families. • The first part collects for the first time in one place, various implications such as Threshold ⇒ Uniquely Realizable ⇒ Degree-Maximal ⇒ Shifted which are equivalent concepts for 2-families (= simple graphs), but strict implications for kfamilies(More)
We introduce the notion of a scheduling problem which is a boolean function S over atomic formulas of the form xi ≤ xj . Considering the xi as jobs to be performed, an integer assignment satisfying S schedules the jobs subject to the constraints of the atomic formulas. The scheduling counting function counts the number of solutions to S. We prove that this(More)
In this paper we show results on the combinatorial properties of shifted simplicial complexes. We prove two intrinsic characterization theorems for this class. The first theorem is in terms of a generalized vicinal preorder. It is shown that a complex is shifted if and only if the preorder is total. Building on this we characterize obstructions to(More)