  We compute the connected Heegaard Floer homology (defined by Hendricks, Hom,
and Lidman) for a large class of 3-manifolds, including all linear combinations
of Seifert fibered homology spheres. We show that for such manifolds, the
connected Floer homology completely determines the local equivalence class of
the associated $\iota$-complex. Some identities relating the rank of the
connected Floer homology to the Rokhlin invariant and the Neumann-Siebenmann
invariant are also derived. Our computations are based on combinatorial models
inspired by the work of N\'emethi on lattice homology.
