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

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