User blog comment:DrCeasium/Hyperfactorial array notation: Analysis part 3/@comment-5529393-20130530121945/@comment-7484840-20130530151109

It duplicates it by k-brackets being both level-k Omega and level-k psi/theta function equivalents. This may seem a bit strange (because it is), but look at the level-1 brackets: they can be equivalents of Omega_0 = 1 ([1] = n), and of the level 0 psi functions (higher structures within type 1 brackets can be represented only using the psi_0 function with the type-1 Omegas) In the same way, the type-2 brackets can be equivalents of Omega_1 (same effect as [_2 1], but the higher structures can only be represented using the psi_1 function of Omega_2 structures. If you like, you could see the brackets themselves as the Omegas, and the structure inside them as the function, so it's sort of inside out in a strange and slightly confusing way.

Another feature of the collapsing functions is that without the psi_0 on the outside, the higher psi_k functions cannot be used with the FGH, because you would end up with f_{cardinal}(n), which doesn't work. With my functions, The same thing applies: n![_2 1] = nothing nested inside itself n times?!?