User blog:GamesFan2000/GamesFan's Arrow Notation System, or Explosively-Powerful Arrow Notation

Part 1: Up-Arrows
1^a = 1

a^1 = aa

a^b = a b+1 length power tower of a’s, for all a and all b

3^3 = (in Knuth up-arrow notation) 3^^4 = 3^3^27

a^b^c = a c+1 length power tower of a’s up-arrowed by a c+1 length power tower of b’s

2^2^2 = 2^16

a^b^c^d = a d+1 length power tower of a’s up-arrowed by a d+1 length power tower of b’s up-arrowed by a d+1 length power tower of c’s

a^^1 = a^a^a

a^^b = a b+2 length single up-arrow chain of a’s

3^^3 = 3^3^3^3^3

a^^b^^c = a c+2 length single up-arrow chain of a’s double up-arrowed by a c+2 length single up-arrow chain of b’s

4^^4^^4 = (4^4^4^4^4^4) ^^ (4^4^4^4^4^4)

a^^^1 = a^^a^^a^^a

a^^^b = a b+3 length double up-arrow chain of a’s

a^^^^b = a b+4 length triple up-arrow chain of a’s

F(n) = an n-length chain of n’s with n arrows between each n

Part 2: Right-Arrows
a>1 = a^^^a^^^a

a>b = a b+2 length chain of a’s with b+2 up-arrows between each a

4>4 = 4^^^^^^4^^^^^^4^^^^^^4

a>b>c = a c+2 length chain of a’s with c+2 up-arrows between each a right-arrowed by a c+2 length chain of b’s with c+2 up-arrows between each b

a>>1 = a>a>a>a>a

a>>b = a b+4 length single right-arrow chain of a’s

a>>>b = a b+6 length double right-arrow chain of a’s

F(n) = an n-length chain of n’s with n right-arrows between each n

Part 3: Combinatorial Arrow Notation
a^>1 = a>a^a^a^a^aa>a^a^a^a^aa>a^a^a^a^aa>a^a^a^a^aa

a^>b = a b+4 length chain of a’s with (b+4 length single up-arrow chain of a’s) right-arrows between each a

a>^b = a b+8 length chain of a’s with (b+8 length chain of a’s with (b+8 length single up-arrow chain of a’s) right-arrows between each a) individual up-right-arrow combos between each a, individual combos represented as ^>, ^>, … Solve them by using the same rule for single combos, removing one of the combos, and repeating this for each new solution until all of the combos are gone.

a^^>b = a b+12 length chain of a’s with (b+12 length chain of a’s with (b+12 length chain of a’s with (b+12 length single up-arrow chain of a’s) right-arrows between each a) individual up-right-arrow combos between each a) individual right-up-arrow combos between each a

The pattern for these is that the chains increase by 4 each time you move up a level if b stays the same, the level previous to you is the arrow combo used, and the equation for the previous level is subscripted to the arrow combo in use for the main expression.

Here’s a list of other combinatorial equations and symbols, in order of power:

a^>^b, a^>>b, a>^^b, a>^>b, a>>^b, ^^^>, ^^>^, ^^>>, ^>^^, ^>^>, ^>>^, ^>>>, >^^^, >^^>, >^>^, >^>>, >>^^, >>^>, >>>^, ^^^^>, ^^^>^, ….

F(n) = an n-length chain of n’s with n combos of n-level between each n, where level 1 is ^>

Before going further, zeroes and negatives aren’t legal for this notation system

Part 4: Left-Arrows
a<1 = a^>, ^>, …^>a…, a

a, ^^<, ^><, >^<, >><, ^<^, ><^, ^<>, ><>, ^<<, ><<, <^^, <^>, <>^, <>>, <^<, <><, <<^, <<>, …

F(n) = an n-length chain of n’s with n n-level EC combos between each n