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 $R$-submodularity 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.