The Ehrhart polynomial of a matroid specializes to the beta invariant

TL;DR

The paper uncovers a direct link between the matroid beta-invariant and the Ehrhart polynomial of the matroid base polytope. It proves that for a loopless and coloopless matroid $\mathsf{M}$ of rank $k$ on $n$ elements, the linear coefficient of $\operatorname{ehr}(\mathsf{M},t-1)$ equals $\beta(\mathsf{M})$ up to the normalization $1/\bigl((n-1)\binom{n-2}{k-1}\bigr)$, i.e. $\left[ t^1 \right]\operatorname{ehr}(\mathsf{M},t-1) = \dfrac{\beta(\mathsf{M})}{(n-1)\binom{n-2}{k-1}}$. The authors establish this by decomposing matroids into sums of snake matroids via a valuative framework, showing that both $\operatorname{ehr}$ and $\beta$ respect this decomposition, and proving a uniform linear-term value for connected snake matroids using their associated fence posets and order polytopes. This yields a lattice-point counting interpretation for $\beta$ and explains a new positivity property of matroid Ehrhart polynomials, while clarifying why the result relies on the loopless/coloopless hypothesis and connects to broader invariants like the $g$- and $\omega$-polynomials. The work also situates the result within the landscape of valuative invariants and polyhedral interpretations of matroid theory.

Abstract

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.

Figures (2)

• Figure 1: The paths $L = \text{EEENENNEN}$ and $U = \text{NEENENNEE}$ for $k = 4$ and $n = 9$. The region between them is shaded.
• Figure 2: The skew shape $\lambda / \mu = (5,4,4,3) / (3,3,2,0)$ and its cell poset $P_{\lambda/\mu}$. Since $\lambda / \mu$ is a border strip, $P_{\lambda/\mu}$ is a fence poset.

Theorems & Definitions (11)

• Theorem 1.1
• Theorem 2.1: ferroni-schroter
• Proposition 2.2
• proof
• Remark 2.3
• Lemma 2.4
• proof
• proof : Proof of Theorem \ref{['thm:main']}
• Remark 2.5
• Example 2.6