The Omega Invariant of a Matroid

Abstract The third author introduced the $g$-polynomial $g_{M}(t)$ of a matroid, a covaluative matroid statistic that 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 $\omega (N)$ is nonnegative for every minor $N$ of $M$, then all the coefficients of $g_{M}(t)$ are nonnegative.