User blog:DrCeasium/Continued hyperfactorial array notation

In the last post, I claimed that I had a challenger to Jonathan Bowers' origional array notation. Well, here it is:

There has been one change since my last post, and that is that now [n] means $$\uarr^n$$, not $$\uarr^{n-2}$$

It is another array notation (as can be guessed from the name). Similarly to the origional array notation by Jonathan Bowers, it has several elemants seperated by commas. It is written as a![b,c,d,...], with as many letters inside the bracket as you want. It is defined by three rules. If rule 1 applies, follow it, if not, if rule 2 applies do it, and if neither of them apply do rule 3. The part of the equation you work on at any one time is the part inside the most sets of brackets (for example in 10![4,[7,[9],4],3,[4,[7,[7,6]],3]], you would expand the [7,6] as this is the part inside the most sets of brackets. If there are two parts drawing over which is inside the most brackets, just choose one, as it should make no difference to the outcome. The rules are:

For $$x_k\in(N)$$ and $$x_k\ne0$$ for $$k\in(N)$$ and $$k\ne0$$, Note 1: if a 1 occurs anywhere inside a set of square brackets, everything after and including the 1 in that set of brackets is irrelivent and can be removed, (for example 5![6,1,4,7] = 5![6]), because 1![x]![y]!... = 1.
 * 1) If $$x_k = 1$$, remove it. (Remove all trailing 1's)
 * 2) If $$k = 2, x_1![x_2] = x_1\uarr^{x_2}(x_1-1)\uarr^{x_2}(x_2-1)\uarr^{x_2}\dots\uarr^{x_2}2\uarr^{x_2}1$$
 * 3) Else, $$x_1![x_2,x_3,x_4,\dots,x_{k-1},x_k] = x_1![x_2,x_3,x_4,\dots,x_{k-2},(x_{k-1}![x_2,x_3,x_4,$$$$\dots,x_{k-1},x_k-1]![x_2,x_3,x_4,\dots,x_{k-1},x_k-2]!\dots![x_2,x_3,x_4,\dots,x_{k-1},1]),x_k-1]$$

Note 2: 2![x,y,z...] = 2, because $$2\uarr^x1 = 1$$

The definitions of these hyperfactorial arrays may seem a little daunting, but is not that difficult, these are a few examples to demonstrate it: The major problem with these, other than their complexity, is that it is in a completely new direction, so it is very difficult to draw links to other numbers or compare them. You could see this as a good thing, in that it is exploring new directions, but it can make the following pretty difficult with other numbers (mostly the ones involving higher starting numbers):
 * 3![3,2] = 3![3![3,1],1] (using rule 3) = 3![3![3]] (using rule 1) = $$3![3^{2^1}]$$ (using rule 2) = 3![9] = $$3\uarr^92\uarr^91$$ (using rule 2) = $$3\uarr^83$$, which is pretty big.
 * 4![3,3] (this not seem much different, but really is) = 4![3![3,2]![3],2] = 4![3![3![3]]![3],2] = 4![3![9]![3],2] = $$4![3\uarr^83![3],2]$$ = $$4![(3\uarr^83![3])![3\uarr^83![3]]]$$ = $$4\uarr^{((3\uarr^83![3])\uarr^{(3\uarr^83![3])}((3\uarr^83![3])-1)\uarr^{(3\uarr^83![3])}\dots3\uarr^{(3\uarr^83![3])}2)}3\uarr^{((3\uarr^83![3])\uarr^{(3\uarr^83![3])}((3\uarr^83![3])-1)\uarr^{(3\uarr^83![3])}\dots3\uarr^{(3\uarr^83![3])}2)}2$$ ... that's.... quite big. On closer inspection, this is easily far larger than g(2) in the Graham's number sequence.
 * 5![5,5,5,5] = 5![5,5,(5![5,5,5,4]![5,5,5,3]![5,5,5,2]![5,5,5]),4] = 5![5,5,(5![5,5,(5![5,5,5,3]![5,5,5,2]![5,5,5]),3]![5,5,(5![5,5,5,2]![5,5,5]),2]![5,5,5![5,5,5]]![5,(5![5,5,4]![5,5,3]![5,5,2]![5,5]),4]),4] =.... I think you probably get the picture.

A common benchmark for big numbers is Graham's number. I have already reffered to some of the terms in its sequence as comparison. These are some comparisons i was able to draw that because $$3\uarr^n2 = 3\uarr^{n-1}3$$, and 3![n] = $$3\uarr^n2$$, and $$g(n) = 3\uarr^{g(n-1)}3$$, it follows that $$3![g(n)+1] = g(n+1)$$. Therefore, Graham's number itself, or g(64) = 3![3![...3![$$3\uarr^43$$+1]...+1]+1], with 63 sets of brackets. This could be greatly overpowered by 3![3,64], as this would contain the necessary brackets and a lot, lot more. In BEAF, Graham's number is roughly {3,65,1,2}, therefore, because of the relative sizes of the representations (or overpowerings) of Graham's number, Hyperfactorial Array Notation is more powerful than even the mighty BEAF, however it lacks the ability to be hugely expanded, which BEAF does very well (or at least it lacks it a the moment ;) ). Don't think for a second this is the end though, as I have Extended Hyperfactorial Array Notation building up right now (or built if its been a long time since I published this). Soon I will also post a list of named numbers formed from this notation. If anyone has any ideas for names, feel free to post them as a comment.