Whitney numbers of the second kind of finite modular lattices

Abstract

Let L be any finite modular lattice. We prove that each interval Jz L has a symmetric and unimodal sequence of Whitney numbers of the second kind iff L may be represented as a direct product of primary q-lattices. We use the term “primary” in the classical sense of Jbnsson and Monk [7], while the term “q-lattice” is from Stanley [lo]. Actually, in the “if” part of the proof, we prove that, upon denoting by n and 1 the dimension and the rank of J, the sequence of the Whitney numbers of J is symmetric and unimodal of the form