By a fundamental result of Giroux, contact structures on 3-manifolds may be described via their open books decomposition. A contact manifold can arise as a boundary of a Stein domain if and only if it has a compatible open book whose monodromy is a product of positive Dehn twists. In principle, one has to examine all compatible open books to detect Stein fillings. However, a theorem of Wendl says that if a compatible open book has planar pages, all Stein fillings are compatible with the given open books. To apply this theorem, we develop combinatorial techniques to study positive monodromy factorizations in the planar case. As a result, we can classify symplectic fillings for all contact structures on $L(p,1)$, and detect non-fillability of certain contact structures on Seifert fibered spaces. (Joint with Jeremy Van Horn-Morris.)