Whitney number inequalities for geometric lattices

Abstract:

Let $L$ be a finite geometric lattice of rank $r$, and for $i = 0,1, \cdots ,r$, let ${W_i}$ denote the number of elements of $L$ with rank $i$. For $1 \leq k \leq r - 2$, we have ${W_1} + {W_2} + \cdots + {W_k} \leq {W_{r - k}} + \cdots + {W_{r - 2}} + {W_{r - 1}}$ with equality if and only if the lattice $L$ is modular. We give two further results concerning matchings of lattice elements of rank $\leq k$ into those of rank $\geq r - k$, and observe that a middle term can be interpolated in the above inequality.