User blog comment:DrCeasium/Hyperfactorial array notation: Analysis part 2/@comment-5529393-20130527105642/@comment-5529393-20130530120745

[@m1... m1@]w/ [@m2... m2@]w/... [@mk... mk@]w/ [q @]@ =  [@m1... m1@]w/ [@m2... m2@]w/...( [@mk... mk@]w/q)w/ [1 @]@.

To make sure I have it right, the ...'s that are in arrays are part of the notation, while the ... outside the arrays is just shorthand for a sequence of arrays, right?

I take it that [@mk... mk@]w/q becomes an array with q mk 's? If so, you need to add that to your definitions.

Unfortunately, this definition does not expand your arrays much at all. It just takes the first entry in the array on the right, and makes that many arrays in the array on the left. For example, if we had

[2...2] w/ [2...2] w/ [2...2] w/ [2...2] w/ [2...2]

we would just get

[2, 2] w/ [1, 2] w/ [1, 2] w/ [1, 2] w/ [1, 2]

You would then need to define what w/ means when there isn't a ... on the left, but in any case it doesn't look like it's anything big.

[@11(b)m@2] = ( [@1(b-1)...(b-1)2(b)m-1@2]w/ [@1(b-1)...(b-1)2(b)m-1@2] w/ [@1(b-1)...(b-1)2(b)m-1@2]...... w/ [@1(b-1)...(b-1)2(b)m-1@2]w/[1])w/[1]

You need to define what happens when b is 0.

As for [1(1)2], we get ([1(0)...(0)2(1)1] w/ [1(0)...(0)2(1)1] w/ [1(0)...(0)2(1)1] ..... [1(0)...(0)2(1)1]w/[1]) w/[1]. I assume the ..... is handled by the outermost w/[1], and as far as I can tell this just means there are n arrays in the sequence. Then the arrays will be resolved using the relatively weak rule above, so you won't even get to \(\phi(\omega) + 1\) - it'll be less.