User blog comment:Mh314159/new YIP notation/@comment-39585023-20190708030201

A partial recursion of {2}0(3) until the appearance of all {1}1 functions. Of course, this is not close to [2], which is {[1]}[1]([1])  -- see previous post about [1].

\[ \{ 2 \}_0 (1) \] \[ \{ 1 \}_{f_1 (1)} (1) \]

\[ \{ 1 \}_{f_0 (2)} (1) \] \[ \{ 1 \}_3 (1) \] \[ \{ 1 \}_2 (4) \] \[ \{ 1 \}_1^{\{ 1 \}_2 (3)} (\{ 1 \}_2 (3)) \] \[ \{ 1 \}_1^{\{ 1 \}_1^{\{ 1 \}_2 (2)} (\{ 1 \}_2 (2))} (\{ 1 \}_2 (3)) \] \[ \{ 1 \}_1^{\{ 1 \}_1^{\{ 1 \}_1^{\{ 1 \}_2 (1)} (\{ 1 \}_2 (1))} (\{ 1 \}_2  (2))} (\{ 1 \}_2 (3)) \] \[ \{ 1 \}_1^{\{ 1 \}_1^{\{ 1 \}_1^{\{ 1 \}_1 (3)} (\{ 1 \}_1 (3))} (\{ 1 \}_2   (2))} (\{ 1 \}_2 (3)) \]