  We introduce a notion of complexity for Sefiert homology spheres by
establishing a correspondence between lattice point counting in tethrahedra and
the Heegaard-Floer homology. This complexity turns out to be equivalent to a
version of Casson invariant and it is monotone under a natural partial order in
the set of Seifert homology spheres. Using this interpretation we prove that
there are finitely many Seifert homology spheres with prescribed Heegaard-Floer
homology. As an application, we characterize L-spaces and weakly elliptic
manifolds among Seifert homology spheres. Also, we list all the Seifert
homology spheres up to complexity two.
