Whitney numbers of some geometric lattices (Q1317449)

The authors study sequences of supersolvable geometric lattices \(L_ 1,L_ 2,L_ 3,\dots\) which are uniform in the sense that every upper interval of \(\text{rank }k\) in \(L_ n\) is isomorphic to \(L_ k\). They derive simple recursions for the Whitney numbers of the lattices \(L_ n\). Writing these in terms of symmetric functions lead to a log-concavity proof in a case with sufficiently many modular coatoms. This yields a common lattice theoretic framework for the known results in the special case where all \(L_ n\) are modularly complemented; here a complete classification of the uniform sequences follows from \textit{J. Kahn} and \textit{J. P. S. Kung} [Eur. J. Comb. 7, 243-248 (1986; Zbl 0614.05018)]. It is an open problem whether other examples exist.