Linear relations for Tutte polynomials of semimatroids

Abstract

We establish a family of linear relations for the coefficients of the Tutte polynomial of a semimatroid. We also introduce ranked central sets, a mild generalization of semimatroids, and prove that the same identities hold in this broader setting. The proof relies on the specialization $x=\frac{z}{z-1}$, $y=z$ along the hyperbola $(x-1)(y-1)=1$; under this specialization, the edge subgraph expansion over central sets reduces to the $f$-polynomial of the central set complex. As applications, we obtain explicit formulas for Tutte polynomials of affine hyperplane arrangements and for balanced Tutte polynomials of biased graphs, whose central sets are central subarrangements and balanced edge sets, respectively. Our results recover the generalized Brylawski's identities for Tutte polynomials of ranked sets, matroids and graphs.