User blog comment:DrCeasium/new hyperfactorial array notation/@comment-5529393-20130416153852/@comment-7484840-20130416192208

The arrays in my notation, at least linear arrays, work like ordinals. This means that in an array [a,b,c,d...] is equivalent to an ordinal as part of an FGH function. [] is equivalent to w, and within an array, the first entry is roughly equivalent to something added to w, eg [3] ~ w + 3 (actually w + 2, because addition has a default of 0). The second entry is equivalent to something multiplied by w, eg [1,3] ~ w.3 (the default of multiplication is 1), the third is equivalent to the exponent, the fourth the tetronent etc. If an array is nested within these, they behave in a very similar way to ordinals being added to or multiplied by another ordinal polynomial. eg [[[2,5],1,8],[3,7],6] ~ ((w^6).(w.7+2))+(w^8)+(w.5+2). The main difference is that with ordinals, you don't normally need brackets when adding, even though it has to evaluate from right to left. The n! is equivalent to the f and (n) in the FGH. That gives a final limit of linear arrays as w^...^w with w up arrows, whatever that is in normal ordinals.

This means that whenever there is an array where you want there to be a number, just use the first entry of that array as the number, just as with an ordinal in the FGH you would use the thing added to it. If it is a complex array with a main entry of 1, treat that sub-array as a full array and work on it until it has a main entry > 1, or just completely decomposes into a number (all the entries are 1).

This ordinal-ness means that nesting arrays is far more powerful than normal, for example:

n![1,1,2] ~ f_{w^2}(n), but n![1,1,[1,1,2]] ~ f_{w^{w^2}}(n) because the arrays work like ordinals in the FGH, so nesting them becomes a lot, lot more powerful.