  We present relations in the mapping class monoid of $S_{0,0}^n$ between
products of boundary parallel twists and those involving only non boundary
parallel twists. These are of particular interest because each element gives an
open book decomposition of a contact 3-manifold, and different factorizations
of the same mapping class give potentially distinct symplectic fillings with
diffeomorphic boundaries. We apply these results in order to compute the Euler
characteristics of fillings resulting from different factorizations, present
plumbing graphs for the fillings given by products of boundary parallel twists,
and provide sharp bounds on the Euler characteristics for fillings arising from
such relations.
