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

The Tutte Polynomial of a Morphism of Matroids 5. Derivatives as Generating Functions of Tutte Activities

We show that in an ordered matroid the partial derivative \partial^{p+q}t/\partialx^p\partialyq of the Tutte polynomial is p!q! times the generating function of activities of subsets with corank p and nullity q. More generally, this property holds for the 3-variable Tutte polynomial of a matroid perspective.Comment: 28 pages, 3 figures, 5 table