Transversal matroids and the half plane property

We focus on checking the validity of the half-plane property on two prominent classes of transversal matroids, namely lattice path matroids and bicircular matroids. We show that lattice path matroids satisfy the half-plane property. Subsequently, we show an explicit example of a bicircular matroid that is not a positroid and discuss the negative correlation properties of bases of transversal matroids. We prove that sparse paving matroids do not satisfy the Rayleigh property, which helps us gain new perspectives about conjectures on negative correlation in basis elements of matroids in general.