User blog:GamesFan2000/Hyper-Extended Factorial Notation (Part 1)

Hyper-Extended Factorial Notation is a set of extensions to ordinary factorials. There will be many different extensions shown in this post. This will also obsolete my earlier extension of factorials. The first part features good ol' regular factorials, multi-factorials, chained factorials and multi-factorials, super-factorials, multi-super-factorials/mixed multi-factorials(grrr...), and chained super-factorials and multi-super-factorials.

Factorials
Since I probably wouldn't hear the end of it otherwise, I will explain regular factorials. Factorials are an extension of multiplication where every non-zero positive integer less than or equal to n is multiplied together. Since 1 does nothing to the total, you can just ignore it.

n! = 1*2*3...*(n-1)*n or ((n-1)!)*n

3! = 2*3 = 6    4! = 6*4 = 24    5! = 120    6! = 720

F(n)'s growth rate is in-between subscripts 2 and 3 on the FGH for factorials.

Multi-Factorials
The first extension of regular factorials will be the multi-factorials.

n!! = ((...((n!)!)...!)!)!

Take n and factorialize it. Factorialize the answer of n!. Repeat this n! times.

2!! = 2    3!! = (((((3!)!)!)!)!)!

n!!! = ((..((n!!)!!)...!!)!!)!!

Take n and double-factorialize it. Double-factorialize the answer of n!!. Repeat this n!! times.

Chained Factorials
Now we move on to chained factorials.

n!a = ((...((n!!!...!!!)!!!...!!!)...!!!...!!!)!!!...!!!)!!!...!!!

Create an a-length set of (n*a)-length multi-factorials of n.

3!3 = ((3!!!!!!!!!)!!!!!!!!!)!!!!!!!!!

n!a!b = n!(((...(((a!!!...!!!)!!!...!!!)!!!...!!!)...!!!...!!!)!!!...!!!)!!!...!!!)!!!...!!!

Chained Multi-Factorials
We now do the next logical step and use multi-factorials in the chains.

n!!a = n!n!n...!n!n!n

Create an (n!a)-length single-factorial chain of n's.

3!!3 = 3!3!3...!3!3!3  (3!3) threes

n!!a!!b = n!!(a!a!a...!a!a!a)

n!!!a = n!!n!!n...!!n!!n!!n

Create an (n!!a)-length double-factorial chain of n's.

n!!!!a = an (n!!!a)-length triple-factorial chain of n's.

Super-Factorials
Now we move on to super-factorials, represented by the symbol ¡.

n¡ = n!!!...!!!n!!!...!!!n...!!!...!!!n

Create an (n^n)-length chain of (n^n)-length multi-factorials of n.

3¡ = 3!!!!!!!!!3!!!!!!!!!3!!!!!!!!!3!!!!!!!!!3!!!!!!!!!3!!!!!!!!!3!!!!!!!!!3!!!!!!!!!3

Mixed Multi-Factorials/Multi-Super-Factorials
Now, by adding the new symbol ¡, this leads to the question of what would happen should both ! and ¡ be used. This caused me grief in the original Extended Factorial Notation, so now it's time to silence the nitpickers. To shut people up about it, here are the mixed multi-factorials, or multi-super-factorials.

n!¡ = ((..((n¡)¡)...¡)¡)¡

Take n and super-factorialize it. Super-factorialize the answer of n¡. Repeat this n¡ times.

n¡! = ((...((n!¡)!¡)...!¡)!¡)!¡

Before I explain this, I will mention that same-length mixed multi-factorials are separated by tiers. You move up the tiers by this logic: Look at the last ¡. If it is the last symbol in the immediate expression, delete the last ! before it, move it into the empty spot, and replace all spaces after the new ¡ with !'s. Otherwise, turn the last ! after the ¡ into a new ¡.

Back to n¡!. Take n and tier-1-double-super-factorialize it. Tier-1-double-super-factorialize the answer to n!¡. Repeat this n!¡ times.

By now, the pattern in this should be obvious.

n¡¡ = an (n¡!)-length repeated set of ¡!'s of n

n!!¡ = an (n¡¡)-length repeated set of ¡¡'s of n

This set is much longer than the ordinary multi-factorials because of the nature of having multiple symbols in the expression.

Chained Super-Factorials
Having more than one type of same-length multi-super-factorials makes chaining super-factorials them slightly more complicated, but if you understand my rules on determing the tiers, they shouldn't be too much of a hassle. The terminology will be changed in that n-tiered multi-super-factorials are used instead of n-length in the definitions.

I can't really show a general expression of n¡a without getting nitpicked at, so I'll just be explaining it.

Create an (n^^a)-length set of (n^^a)-tiered multi-super-factorials of n.

3¡3 = (You wish I would show the full expression, but 3^27 repetitions of 3^27-tiered multi-super-factorials...I'll pass.)

3¡3¡3 = (This is even crazier than the above expression.)

Chained Mixed Multi-Factorials/Chained Multi-Super-Factorials
As always, if you can do it in a single, you can do it in a multi. Here are the chained multi-super-factorials.

n!¡a = an (n¡a)-length super-factorial chain of n's

n¡!a = an (n!¡a)-length !¡-chain of n's

n¡¡a = an (n¡!a)-length ¡!-chain of n's

n!!¡a = an (n¡¡a)-length ¡¡-chain of n's

All right, that's the first part of HEFN completed. I would appreciate if people tried to solve the growth rates, because I suck at FGH.