On the cohomology of a class of nilpotent Lie algebras

Let g denote a finite dimensional nilpotent Lie algebra over C containing an Abelian ideal a of codimension 1, with z is an element of g\a. We give a combinatorial description of the Betti numbers of g in term of the Jordan decomposition [GRAPHICS] induced by ad(z)\(a). As an application we prove that the filiform-nilpotent Lie algebras arising in the case t = 1 have unimodal Betti numbers.