User blog comment:DrCeasium/Hyperfactorial array notation: Analysis part 3/@comment-5529393-20130530121945/@comment-7484840-20130530164634

A type-2 bracket, with only a main entry of 1, is defined as nesting the next type-1 bracket out, and everything in it, in the position that the type-2 bracket used to be in, n times, in the same way that an Omega is replaced by the entire function and all it's contents nested in it's place n times. If the type-2 bracket has more than a main entry of 1, then it is not just n type-1 brackets, it will evaluate to produce type-2 brackets with only a main entry of 1, in the same way that larger Omega structures will produce Omegas that can use their evaluating rule.

My notation is equivalent to the 'strong' psi function, because there are higher-psi equivalents, because when there has to be a type-(k-1) bracket for a type-k bracket to work by definition (of course, type-1 brackets are an exception). However, like in the (strong) psi function, these can be added in if they are not there.

Basically, like the psi function, each level of brackets in my notation 'collapses' the structure inside it into a much larger structure in a lower type of bracket.