Whitney Numbers of Partial Dowling Lattices

The Dowling lattice $Q_n(\mathfrak{G})$, $\mathfrak{G}$ 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 $\mathfrak{G}\cdot K_n^{(V)}$. Its Whitney numbers of the first kind enter into several important formulas. Ravagnani suggested and partially proved that these numbers of $Q_n(\mathfrak{G})$ and higher-weight generalizations are polynomial functions of $\mathrm{\mid }\mathfrak{G}\mathrm{\mid }$. We give a simple proof for $Q_n(\mathfrak{G})$ and its generalization to a wider class of gain graphs and biased graphs, and we determine the degrees and coefficients of the polynomials.