User blog:GamesFan2000/Powerful Array Notation (Part 1)

Powerful Array Notation is yet another array notation from me. It works differently this time, of course.

Terminology
Array: a structure in math that can display entries in a set

Array notation: Array notation was invented by Jonathan Bowers, and usually takes the form of {a, b, c, …}. Bowers, Chris Bird, and hyp cos all have really good generalizations of this type of notation.

Pilot: represented by variable a, the pilot is the first entry of the array

Co-pilot: represented by variable b, the co-pilot is the second entry of the array

Passengers: represented by other variables, the passengers are all entries that come after the co-pilot

Chained arrow notation: the base function for my notation, invented by John Conway. It takes the form of a→b→c…

Knuth’s up-arrow notation: the easiest way to express operations beyond exponentiation, invented by Donald Knuth. It takes the form of a^^^^…^^^^b.

Rules: Two-Entry Arrays
Rule 1: (a) = a

Rule 2: (1, b, …) = 1, (a, 1) = a

Rule 3: (a, b) = a→a→a→a…b a’s…a

Examples
(2, 2) = 2→2 = 4

(3, 2) = 3→3 = 27

a→b = a^b

(4, 2) = 256

(2, b) = 4

(3, 3) = 3→3→3 = 3^^^3

(4, 3) = 4→4→4 = 4^^^^4

a→b→c = a^^^…c ^’s…^^^b

(3, 4) = 3→3→3→3 = 3→3→(3→3→2→3)→2 = 3→3→(3→3→(3→3→1→3)→2)→2 =

3→3→(3→3→27→2)→2 = 3→3→(3→3→(3→3→26→2))→2 = 3→3→(3→3→(3→3→(3→3→25→2)))→2… (This number is MUCH larger than Graham’s number.)

(4, 4) = 4→4→4→4

(3, 5) = 3→3→3→3→3

(n, n) = n→n→n→n→n…n n’s…→n

Rules: Three-Entry Arrays and larger
Rule 4: (a, b, …1) = (a, b, …)

Rule 5: (a, b, c) = ((a, b, c-1), (a, b, c-1), c-1)

Rule 6: ((a, b, c), d) = (((a, b, c-1), (a, b, c-1), c-1), d)

Rule 7: (a, b, c, …d) = ((a, b, c, …d-1), (a, b, c, …d-1), (a, b, c, …d-1), …d-1)

Examples
(3, 3, 2) = ((3, 3), (3, 3)) = (3^^^3, 3^^^3)

(3, 1, 2) = (3, 3)

(3, 3, 3) = ((3, 3, 2), (3, 3, 2), 2) = (((3, 3), (3, 3)), ((3, 3), (3, 3)), 2) = ((3^^^3, 3^^^3), (3^^^3, 3^^^3), 2)

(3, 3, 3, 3) = ((3, 3, 3, 2), (3, 3, 3, 2), (3, 3, 3, 2), 2) = (((3, 3, 3), (3, 3, 3), (3, 3, 3)), ((3, 3, 3), (3, 3, 3), (3, 3, 3)), ((3, 3, 3), (3, 3, 3), (3, 3, 3)), 2)

F(n) = (n, n, n, n, n, …n n’s…n)