User blog comment:DrCeasium/new hyperfactorial array notation/@comment-5529393-20130416153852/@comment-7484840-20130417175502

You are right in your comparison as  n![1,1,1,1,2] = f_epsilon_0(n) and  n![1,1,1;2] = f_epsilon_0(n), however, the 2nd line does grow more powerful than the first, because:

n![1,1,1;2] ~ f_{e_0}(n)

really, we can ignore the centre nest with the active entry set to 1 (I might just remove this rule altogether)

n![1,1,1;3] = n![1,1,[1,1[1,1...[1,1,1;2]...;2];2];2] ~ f_{e_1}(n), and why, well, repeatedly nesting [1,1,1;2] in the exponent equivalent entry is equivalent to (e_0)^(e_0)^(e_0)... = e_1

This applies generally to n![1,1,1;k] =  n![1,1,[1,1[1,1...[1,1,1;k-1]...; k-1] ; k-1 ]; k-1 ], in the same way that  (e_{k-1})^(e_ {k-1} )^(e_ {k-1} )... = e_k, therefore,  n![1,1,1;n] ~ f_{e_w}(n). Not that powerful. The real power in the second row comes from the later entries, starting with the second.

n![1,1,1;1,2] ~  n![1,1,1; [1,1,1; [1,1,1;... [1,1,1;n]... ] ] ]. We know that the 1st entry in the second row is equivalent to the subscript of epsilon (well, it's 2 less), and  [1,1,1;n] ~ e_{n-2}, or e_w. Of course, because these arrays behave like ordinals, nesting works as it does with ordinals. Therefore  [1,1,1; [1,1,1;n] ] is equivalent to e_w in the subscript of epsilon itself, or e_{e_w}. Nesting this n times (as it would for n![1,1,1;1,2]) is equivalent to e_{e_{e_{...e_{e_w}...}}}, or zeta_0. n![1,1,1;1,3] is equivalent to nesting zeta in its subscript n times, or eta_0. This continues in this fashion and generalises to n![1,1,1;1,k] ~ f_{phi_n(0)}(n), or  n![1,1,1;1,n] ~ f_{phi_w(0)}, or the power of linear arrays already.

In a sense, in the second row, the first entry is the subscript of an ordinal below  phi_w(0), and the second entry is its 'type', as in type 1 = e, type 2 = z, and so on. Now what about the 3rd?

The third entry = 2 will nest in the 'ordinal type' entry n times, or   phi_{ phi_{ phi_{... phi_0(0)...} (0) }(0)} (0). This is equal to gamma_0.

It turns out that the definition of the second row is very similar to that of the extended veblen function, because every entry nests n times in the one before it (or after it in the veblen function), and they work just like ordinals.

In summary: nesting is nothing new, but when combined with the ordinal nature of these arrays, it works in a completely different way to a non-ordinal-esque array, not following the usual rules on how iteration usually works.