The Ehrhart polynomial of a matroid specializes to the beta invariant

Key Points

• The linear coefficient of the Ehrhart polynomial relates to the beta invariant of the matroid, revealing deep connections.
• A lattice-point counting formula emerges from this relationship, offering insights into the structure of matroid polytopes.
• The results suggest a positivity property for Ehrhart polynomials that was previously unexpected.
• This work advances our understanding of matroid polytopes, linking combinatorial properties through polynomial evaluation.