The omega invariant of a matroid

The third author introduced the $g$-polynomial $g_M(t)$ of a matroid, a covaluative matroid statistic which is unchanged under series and parallel extension. The $g$-polynomial of a rank $r$ matroid $M$ has the form $g_{1}t+g_2{t}^2+\cdots +g_r{t}^r$. The coefficient $g_1$ is Crapo’s classical $\beta$-invariant. In this paper, we study the coefficient $g_r$, which we term the $\omega$-invariant of $M$. We show that, if $M\mathrm{/}F$ is connected for every proper flat $F$ of $M$, and $\omega (N)$ is nonnegative for every minor $N$ of $M$, then all the coefficients of $g_M(t)$ are nonnegative. We give several simplified versions of Ferroni’s formula for $\omega (M)$, and compute $\omega (M)$ when $r$ or $\mathrm{\mid }E(M)\mathrm{\mid }-2r$ is small.