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We consider the set of alternating paths on a fixed fully packed loop of sizen. 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 theset of alternating paths are nested osculating loops, which we call Dyckislands. Dyck islands can be constructed as a union of lattice Dyck paths, andwe use this structure to give a simple graphical formula for the calculation ofthe inversion number of an alternating sign matrix.

Alternating paths of fully packed loops and inversion number