Year: 2025
Type: article
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.
DOI: 10.1093/imrn/rnaf329
Authors: Alex Fink, Kris Shaw, David E Speyer
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $4,534
OpenAlex ID: W4415599726