On the number of simple arrangements of five double pseudolines

We describe an incremental algorithm to enumerate the isomorphism classes of double pseudoline arrangements. The correction of our algorithm is based on the connectedness under mutations of the spaces of one-extensions of double pseudoline arrangements, proved in this paper. Counting results derived from an implementation of our algorithm are also reported.Comment: 24 pages, 16 figures, 6 table

The Tutte Polynomial of a Morphism of Matroids 6. A Multi-Faceted Counting Formula for Hyperplane Regions and Acyclic Orientations

We show that the 4-variable generating function of certain orientation related parameters of an ordered oriented matroid is the evaluation at (x + u, y+v) of its Tutte polynomial. This evaluation contains as special cases the counting of regions in hyperplane arrangements and of acyclic orientations in graphs. Several new 2-variable expansions of the Tutte polynomial of an oriented matroid follow as corollaries. This result hold more generally for oriented matroid perspectives, with specific special cases the counting of bounded regions in hyperplane arrangements or of bipolar acyclic orientations in graphs. In corollary, we obtain expressions for the partial derivatives of the Tutte polynomial as generating functions of the same orientation parameters.Comment: 23 pages, 2 figures, 3 table

On a Mutation Problem for Oriented Matroids

AbstractFor uniform oriented matroids M with n elements, there is in the realizable case a sharp lower bound Lr(n) for the number mut(M) of mutations of M: Lr(n) =n≤mut(M), see Shannon [17]. Finding a sharp lower bound L(n) ≤mut(M) in the non-realizable case is an open problem for rank d≥ 4. Las Vergnas [9] conjectured that 1 ≤L(n). We study in this article the rank 4 case. Richter-Gebert [11] showed thatL (4 k) ≤ 3 k+ 1 for k≥ 2. We confirm Las Vergnas’ conjecture for n< 13. We show that L(7k+c) ≤ 5 k+c for all integersk≥ 0 and c≥ 4, and we provide a 17 element example with a mutation free element

LR characterization of chirotopes of finite planar families of pairwise disjoint convex bodies

We extend the classical LR characterization of chirotopes of finite planar families of points to chirotopes of finite planar families of pairwise disjoint convex bodies: a map \c{hi} on the set of 3-subsets of a finite set I is a chirotope of finite planar families of pairwise disjoint convex bodies if and only if for every 3-, 4-, and 5-subset J of I the restriction of \c{hi} to the set of 3-subsets of J is a chirotope of finite planar families of pairwise disjoint convex bodies. Our main tool is the polarity map, i.e., the map that assigns to a convex body the set of lines missing its interior, from which we derive the key notion of arrangements of double pseudolines, introduced for the first time in this paper.Comment: 100 pages, 73 figures; accepted manuscript versio