Whitney Numbers of Partial Dowling Lattices

TL;DR

This paper provides a simple proof for the polynomial nature of Whitney numbers of the first kind in Dowling lattices and their generalizations, including determining their degrees and coefficients.

Contribution

It offers a straightforward proof that Whitney numbers are polynomial functions of group order and extends this to broader classes of gain and biased graphs.

Findings

Whitney numbers are polynomial functions of group size

Determined degrees and coefficients of these polynomials

Extended results to wider classes of gain and biased graphs

Abstract

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 $|\mathfrak{G}|$. 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.