The Alpha Function of a Matroid—I. Transversal Matroids

Let n be the nullity function of the matroid G(S). The Mason alpha function is defined on subsets A of S by the recursion α(A)=n(A)?∑_F?Aα(F), the summation being over all flats F strictly contained in A. The alpha function may be viewed as the first difference of the nullity. We study the behavior of a under strong maps, and apply our results to proving Mason's alpha criterion: a matroid is the dual of a transversal matroid if and only if its alpha function is non-negative.