Given a positive factorisation of the identity in the mapping class group of a surface S, we can associate to it a Lefschetz fibration over S^2 with S as a regular fiber. Its total space X is a symplectic 4-manifold, so it is natural to ask what kind of invariants of X can be read off from this construction. I will report on an ongoing joint work with Tim Perutz, aimed at obtaining an explicit formula for counting holomorphic sections of a Lefschetz fibration over the disk, while keeping track of their relative homology classes. This is the first step in our program to give explicit formulas for the Donaldson-Smith invariants of a Lefschetz fibration which, thanks to a combination of results by Usher and Taubes, are equivalent to the SW invariants of X. (This is my dissertation defense)