Analysis uncovers a relationship between Ehrhart polynomials and beta invariants, indicating new positivity properties.
We show that the linear coefficient of the Ehrhart polynomial of a matroid base polytope evaluated at $t-1$ is equal to, up to normalization, the β-invariant of the matroid. This yields a lattice-point counting formula for the β-invariant and establishes a new and unexpected positivity property of Ehrhart polynomials of matroid polytopes.
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Chavez et al. (2025) studied this question.
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