The paper gives a bijective proof of Zaslavsky's level enumeration for hyperplane arrangements via centralization, shows that the counts depend only on the intersection poset, and derives a general characteristic polynomial for geometric semilattices with applications to braid deformations.
Semimatroids and their Tutte polynomials
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abstract
We define and study "semimatroids", a class of objects which abstracts the dependence properties of an affine hyperplane arrangement. We show that geometric semilattices are precisely the posets of flats of semimatroids. We define and investigate the Tutte polynomial of a semimatroid. We prove that it is the universal Tutte-Grothendieck invariant for semimatroids, and we give a combinatorial interpretation for its non-negative coefficients.
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2025 1verdicts
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Region level via centralization for hyperplane arrangements and beyond
The paper gives a bijective proof of Zaslavsky's level enumeration for hyperplane arrangements via centralization, shows that the counts depend only on the intersection poset, and derives a general characteristic polynomial for geometric semilattices with applications to braid deformations.