Whitney number inequalities for geometric lattices

Abstract

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