User blog comment:DrCeasium/Hyperfactorial array notation: Analysis part 3/@comment-5529393-20130530121945/@comment-7484840-20130530154648

type 2 brackets are the equivalent of Omega_1, because (when put inside type 1 brackets or an ordinal collapsing function respectively):

2 1 = $$\alpha\mapsto [\alpha]$$ (which is actually not that big. This is why I put the [_2 1] in the second row in the examples above: to 'kickstart' the function at gamma_0)

$$\psi(\Omega) = \alpha\mapsto\psi(\alpha)$$

Some more examples about the similarities in the way the 2 notations work:

[k 1,1,3] = [k 1,[2 1],2] $$\Omega^3 = \Omega^2\times\Omega$$

[k@[k+11]@] = $$\alpha\mapsto[_k@\alpha@]$$ $$\psi_k(@\Omega_{k+1}@) = \alpha\mapsto\psi_k(@\alpha@)$$

Saying type-2 brackets are just n type-1 brackets is sort of like saying $$\psi(\Omega)$$ is just n psi functions. Once you start putting larger arrays/Omega structures in the brackets, they are a lot, lot more than just n type-1 brackets/functions