Amalgamation over uniform matroids

Uniform Families and Count Matroids

Matroids over a ring

We introduce the notion of a matroid M over a commutative ring R, assigning to every subset of the ground set an R-module according to some axioms. When R is a field, we recover matroids. When R = Z, and when R is a DVR, we get (structures which contain all the data of) quasi-arithmetic matroids, and valuated matroids, respectively. More generally, whenever R is a Dedekind domain, we extend the...

On series-parallel extensions of uniform matroids

This paper gives an excluded-minor characterization of the class of matroids that are series-parallel extensions of uniform matroids.

Total Polynomials of Uniform Oriented Matroids

In this paper we introduce polynomials associated with uniform oriented matroids whose coefficients enumerate cells in the corresponding arrangements. These polynomials are quite useful in the study of many enumeration problems of combinatorial geometry, such as counting faces of polytopes, counting Radon partitions, counting k-sets, and so forth. We also describe some striking equations relati...

Separoids and characterization of linear uniform oriented matroids

In this paper the geometric dimension of an oriented matroid is introduced. It is the minimal euclidian dimension where its separoid (to be defined) can be realized as a family of convex sets. We show that in the uniform case, it is enough to know this invariant to decide if the oriented matroid is linear .