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.