Order polytopes of thin posets and log-concavity of h*-vectors

Does every lattice polytope with the integer decomposition property (IDP) have a unimodal h*-vector? This question, attributed to Stanley, is one in a hierarchy of conjectures involving distributional properties of h*-vectors of lattice polytopes. Even in the case of IDP polytopes whose h*-vectors have explicit combinatorial interpretations — such as the order polytope of naturally labeled posets — this question is wide open. We focus on the subclass of thin posets P and approach Stanley’s question by finding a novel matroidal interpretation of P-Eulerian polynomials in this case. In doing so, we make progress on unimodality and log-concavity conjectures of Stanley and Brenti respectively. We relate this new class of matroids to transversal matroids, lattice path matroids, and positroids. This is joint work with Per Alexandersson.