The paper gives a cone-system construction for Lindelöf scattered P-spaces and shows that a length-ω2 cardinal sequence is realizable between two almost-disjointness bounds, which consistently differ, so no ZFC characterization is known.
Engelking, General Topology, volume 6 of Sigma Series in Pure Mathematics , Heldermann Verlag, Berlin, second edition, 1989
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Cardinal Sequences of Lindel\"of scattered P-spaces
The paper gives a cone-system construction for Lindelöf scattered P-spaces and shows that a length-ω2 cardinal sequence is realizable between two almost-disjointness bounds, which consistently differ, so no ZFC characterization is known.