  We prove a duality theorem for simplicial complexes arising from a
combinatorial construction we define, which applies to the squarefree monomial
complexes for Veronese ideals of projective spaces and weighted projective
spaces. Our theorem yields a formula for the multigraded Betti numbers of these
Veronese ideals in terms of the reduced homology groups of these complexes
which is dual to one given by Bruns and Herzog. We apply this formula in
several ways, including by giving an algorithm for finding the highest syzygy
of such a Veronese ideal.
