Flags and Whitney numbers of matroids (Q1321997)

Let \([X_ i]\) be a saturated chain of flats in a rank-\(r\) simple matroid \(G\) and let \(a_ i\) be the number of points in \(X_ i\) but not in \(X_{i-1}\). We prove that the \(m\)th Whitney number \(w_ m(G)\) of the first kind (defined to be the sum \(\sum\mu(\widehat 0,X)\) over all rank- \(m\) flats \(X\)) is greater than or equal to the coefficient of \(\lambda^{r-m}\) in the polynomial \((\lambda- a_ 1)(\lambda- a_ 2)\cdots (\lambda-a_ r)\). Equality occurs for any \(m\) in the range \(2\leq m\leq r\) if and only if all the flats \(X_ i\) are modular.