User blog:GamesFan2000/Extended Factorial Notation

Extended Factorial Notation
Extended Factorial Notation is the first major set of extensions to regular factorial notation. Within EFN are the ordinary factorials and three extensions to the system. The first extension is Chained Factorial Notation, which is a long line of exclamation points with numbers surrounding them. The second extension is Multi-Factorial Notation, which takes a page from Knuth’s up-arrow notation and uses multiple exclamation points between the numbers. The third extension is Upside-Down Exclamation Point Factorial Notation, which defines how upside-down exclamation points are used.

Section 1.1: Factorial Notation
Before going into the main part of this system, we should define what a factorial is. In math, there are multiple different operators that can be used. They all amount to over-glorified counting, because counting is the most basic operation in math. Addition is repeated counting and multiplication is repeated addition. Multiplication is the operation that factorials are based in. It works as follows:

4×4 = 4+4+4+4 = 16 (the sum of a b length set of as’)

Repeated multiplication is known as exponentiation. However, exponents are beyond the scope of normal factorialization. Factorials are defined as follows:

n! = (n-(n-1)) × (n-(n-2)) × … (n-2) × (n-1) × (n)

Create a set of numbers equal to n that is n long. Right to left from the second-last entry to the first entry, subtract from each number, starting at one for the second-last entry and increasing the subtraction by one each time, such that the sequence is 1, 2, 3, …n. Put multiplication symbols between each number and solve.

1! = 1    2! = 1×2 = 2    3! = 1×2×3 = 6

Since the 1 at the start does nothing, we can just ignore it.

4! = 2×3×4 = 24    5! = 5×4! = 120

n! = n×(n-1)!

6! = 720    7! = 5040    8! = 40320    9! = 362840

Factorials are the base operation of this system, and will be the completely simplified state of every extension to follow.

Section 1.2: Chained Factorial Notation
Now we get into the first extension in this system. A factorial multiplies n by every number less than n. However, that is puny compared to other operations. Even exponentiation is more powerful than ordinary factorials, as exponents are comparable to f3(n) in the fast-growing hierarchy while factorials are only comparable to f2(n). That growth rate leaves much to be desired, so we need to find a way to increase the power. At this point, we ask: what if we use n!n? In this system, we define that as follows:

n!a = (1 × 2 × 3 … × n) × (1 × 2 × 3 … × n) × … (1 × 2 × 3 … × n)

Create an a-length set of n-length sets of n’s. Solve each set like you would in normal factorials, then multiply them with each other.

1!1 = 1    2!2 = 4    2!3 = 8    3!2 = 36    3!3 = 216    4!3 = 13784

n!a!b = n!((1 × 2 … × a) × … (1 × 2 … × a))

2!2!2 = 2!4 = 16    3!3!3 = 3!216

As you can see, n!n > nn. This is true of all n greater than 2. If n = 1 or 2, exponentiation and chained factorialization are equal. Also, !n is an invalid operation in this system if another number isn’t in front of the !.

Section 1.3: Multi-Factorial Notation
We’ve defeated exponents with the first extension of this system, but we are still miniscule compared to the non-mainstream hyper-operations. Take, for instance, tetration. That’s repeated exponentiation. Our chained factorials can’t compete with that. We still have a long way to go. Let’s introduce the second extension of our factorial system. It works as follows:

n!!a = n!n!n!n…!n

The double exclamation point means to create a repeated chain of n’s. The chain will have a length equal to a.

2!!2 = 2!2 = 4    3!!3 = 3!3!3    3!!3!!3 = 3!!(3!3!3)

n!!!a = n!!n!!n…!!n

n!!!!a = n!!!n!!!n…!!!n

You are not restricted to using one kind of multi-factorial for your expression. 3!!!3!!3 is completely valid. Just solve from right to left. That said, we’ve now surpassed the hyper-operators, which are capped at fω(n). Also, n!! isn’t valid.

Section 1.4: Upside-Down Exclamation Point Factorial Notation
We are now ready to add our first new symbol to the system. Remember that multi-factorials are capped at just slightly more powerful than fω(n), which means that we’re in the realm of chained arrow notation at this point. We will now add in the ¡ operator. This works as follows:

n¡ = n!!!...!!!n!!!...!!!n…n

Create an n-length chain of n’s with n !’s between each n.

3¡ = 3!!!3!!!3

n¡a = (n!!!...!!!n…n)!!!...!!!(n!!!...!!!n…n)…(n!!!...!!!n…n)

Create an a-length chain of n-length chains of n’s, with a !’s between each chain and n !’s between each n.

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

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

n¡¡a = n¡n¡n…¡n

Create an a-length chain of n’s with upside-down exclamation points between each n.

3¡¡4 = 3¡3¡3¡3    3¡¡¡4 = 3¡¡3¡¡3¡¡3    3¡¡¡¡4 = 3¡¡¡3¡¡¡3¡¡¡3

With the upside-down exclamation points taken care of, EFN has reached its limits.