Joint work with Riccardo Pedrotti (UT Austin). A finite sequence of simple closed curves on a surface determines a symplectic 4-manifold with boundary, as the total space of a Lefschetz fibration over the disc with the given surface as reference fiber and the curves as vanishing cycles. The monodromy around the boundary of the disc is the composite of Dehn twists. A positive relation in the mapping class group - a composite of Dehn twists isotopic to the identity - gives an example with trivial monodromy, in which case one can complete the 4-manifold to a closed symplectic 4-manifold with a Lefschetz fibration over the 2-sphere. This construction is known to account for (a blow-up of) every closed symplectic 4-manifold. It is natural, then, to ask for a recipe for the Seiberg-Witten invariants of these 4-manifolds in terms of the given sequence of curves. Donaldson-Smith interpreted the SW invariants of a Lefschetz fibration as counts of pseudo-holomorphic multi-sections representing a fixed homology class. We use symplectic Floer theory to obtain a "formula" for the count of pseudo-holomorphic sections (not yet multi-sections), over the disc, in a given homology class (rel boundary). The answer is expressed in terms of domains in the fiber surface.