On the Ehrhart polynomial of minimal matroids. (English) Zbl 1490.05026

Summary: We provide a formula for the Ehrhart polynomial of the connected matroid of size \(n\) and rank \(k\) with the least number of bases, also known as a minimal matroid. We prove that their polytopes are Ehrhart positive and \(h^\ast\)-real-rooted (and hence unimodal). We prove that the operation of circuit-hyperplane relaxation relates minimal matroids and matroid polytopes subdivisions, and also preserves Ehrhart positivity. We state two conjectures: that indeed all matroids are \(h^\ast\)-real-rooted, and that the coefficients of the Ehrhart polynomial of a connected matroid of fixed rank and cardinality are bounded by those of the corresponding minimal matroid and the corresponding uniform matroid.


05B35 Combinatorial aspects of matroids and geometric lattices
52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)
11B73 Bell and Stirling numbers
Full Text: DOI arXiv


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