Abstract
The goal of this note is to highlight applications to graph orientations of some recent results in oriented matroid perspectives, by providing adapted settings and a detailed example. An (oriented) matroid perspective (or morphism, or strong map, or quotient) is an ordered pair of (oriented) matroids satisfying some structural relationship. In the case of graphs, two notable types of perspectives can be considered: graph homomorphisms, and dually embedded graphs on a surface. The Tutte polynomial of such a perspective is a classical polynomial (also called Las Vergnas polynomial in the case of dually embedded graphs), whose coefficients and (some) evaluations are known to count pairs of orientations of certain types. We show how coefficients and (other) evaluations of the polynomial also count pairs of orientations of certain types where some edge orientations are fixed, as well as some equivalence classes of pairs of orientations of certain types. These properties appear when the edge set is linearly ordered.