The Poincare Polynomial of an Arrangement with the Trio Separation Property

Abstract

An arrangement of hyperplanes with a modular element in its intersection lattice has a Poincaré polynomial which factors; this was proven by Stanley in the setting of geometric lattices. This note proves a factorization in the setting of hyperplane arrangements under two conditions which imply a modular element. Two well known reflection arrangements serve as motivation and their Poincaré polynomials are computed using the main theorem of this note.