  We consider the set of alternating paths on a fixed fully packed loop of size
n. This set is in bijection with the set of fully packed loops of size n.
Furthermore, for a special choice of fully packed loop, we demonstrate that the
set of alternating paths are nested osculating loops, which we call Dyck
islands. Dyck islands can be constructed as a union of lattice Dyck paths, and
we use this structure to give a simple graphical formula for the calculation of
the inversion number of an alternating sign matrix.
