  The aim of this paper is to use mapping class group relations to approach the
`geography' problem for Stein fillings of a contact 3-manifold. In particular,
we adapt a formula of Endo and Nagami so as to calculate the signature of such
fillings as a sum of the signatures of basic relations in the monodromy of a
related open book decomposition. We combine this with a theorem of Wendl to
show that for any Stein filling of a contact structure supported by a planar
open book decomposition, the sum of the signature and Euler characteristic
depends only on the contact manifold. This gives a simple obstruction to
planarity, which we interpret in terms of existence of certain configurations
of curves in a factorization of the monodromy. We use these techniques to
demonstrate examples of non-planar structures which cannot be shown non-planar
by previously existing methods.
