whitney numbers of partial dowling lattices

The dowling lattice qn(g), g a finite group, generalizes the geometric lattice generated by all vectors, over a field, with at most two nonzero components. abstractly, it is a fundamental object in the classification of finite matroids. constructively, it is the frame matroid of a certain gain graph known as g·k(v )n . its whitney numbers of the first kind enter into several important formulas. ravagnani suggested and partially proved that these numbers of qn(g) and higher-weight generalizations are polynomial functions of |g|. we give a simple proof for qn(g) and its generalization to a wider class of gain graphs and biased graphs, and we determine the degrees and coefficients of the polynomials.