Combinatorial Inequalities for Geometric Lattices

A geometric lattice is a semimodular point lattice L. The ith Whitney number of Lis the number of elements of rank i in L. The logarithmic concavity conjecture states that Wi(L)2/Wi-1(L)Wi+1(L) ≥ 1 for any finite geometric lattice L. In a finite nondirected graph without loops or double edges, a set of edges is closed if whenever it contains all but one edge of a cycle, it contains the whole cycle. With set containment as the order relation, the closed sets of such a graph form a geometric lattice. It is shown that any such lattice satisfies the first nontrivial case of the logarithmic concavity conjecture. In fact, W2(L)2/W1(L)W3(L) ≥ 3/2 · (W1(L)-1)/(W1(L)-2) · This is a best possible result since equality holds for graphs without cycles. The cut-contraction of a geometric lattice L with respect to a modular cut Q of L is the geometric lattice L - T where T = {x Є L : x Є Q, Ǝq Є Q Э x q}. It is shown that any geometric lattice L can be obtained from the Boolean algebra with W1(L) points by means of a sequence of k = W1(L) - dim(L) cut-contractions.