Flags and Whitney Numbers of Matroids

Let [Xi] be a saturated chain of flats in a rank-r simple matroid G and let ai be the number of points in Xi, but not in Xi − 1. We prove that the mth Whitney number wm(G) of the first kind (defined to be the absolute value of the sum ∑ μ(0̂, X) over all rank-m flats X) is greater than or equal to the coefficient of λr − m in the polynomial (λ − a1)(λ − a2) ··· (λ − ar). Equality occurs for any m in the range 2 ≤ m ≤ r if and only if all the flats Xi, are modular.