Pure patterns of order 2

We provide mutual elementary recursive order isomorphisms between classical ordinal notations, based on Skolem hulling, and notations from pure elementary patterns of resemblance of order 2, showing that the latter characterize the proof-theoretic ordinal 1∞ of the fragment Π11-CA0 of second order number theory, or equivalently the set theory KPℓ0. As a corollary, we prove that Carlson's result on the well-quasi orderedness of respecting forests of order 2 implies transfinite induction up to the ordinal 1∞. We expect that our approach will facilitate analysis of more powerful systems of patterns.