Whitney numbers of some geometric lattices

Abstract

It is a well known fact that the Whitney numbers of a sequence of ranked posets P0, P1, P2,-.of rank 0, l, 2 .... such that every r-ranked filter of P, is isomorphic to Pr are the connection constants between the sequence of powers and the sequence of the characteristic polynomials of the posets [-Dow]. We note here that, if the posets are indeed supersolvable geometric lattices, then the sequence of characteristic polynomials is persistent in the sense of [Dam2]. Thus, in Sections 3 and 4, we will be able to transfer the two term recursion and the explicit formula for connection constants [Daml] as well as several log-concavity properties of symmetric functions [Sag2] to the Whitney numbers of those lattices. Moreover, since the roots of the characteristic polynomial of a supersolvable lattice have been given a nice combinatorial meaning [Sta2], we can also give a simple semantics for those formulas. As a consequence of our results, we obtain (Section 5) unifying proofs of several properties enjoyed by Whitney numbers of Boolean algebras, subspace lattices, partition lattices, and Dowling lattices. It turns out that these lattices form the only infinite sequences of modularly complemented geometric lattices satisfying the conditions mentioned at the beginning.