An Extension of Matroid Rank Submodularity and the Z-Rayleigh Property

We define an extension of matroid rank submodularity called R-submodularity, and introduce a minor-closed class of matroids called extended submodular matroids that are well-behaved with respect to R-submodularity. We apply R-submodularity to study a class of matroids with negatively correlated multivariate Tutte polynomials called the Z-Rayleigh matroids. First, we show that the class of extended submodular matroids are Z-Rayleigh. Second, we characterize a minor-minimal non-Z-Rayleigh matroid using its R-submodular properties. Lastly, we use Rsubmodularity to show that the Fano and non-Fano matroids (neither of which is extended submodular) are Z-Rayleigh, thus giving the first known examples of Z-Rayleigh matroids without the half-plane property.