User blog:GamesFan2000/Entry Swap Array Notation

Christmas is all about the giving spirit, and that’s no different in googology. As an early Christmas present, here’s another array notation for you. I call this the Entry Swap Array Notation.

Terminology
Array – An array is a string of integers enclosed by a pair of brackets. These take the form of {a, b, c, d, ...}.

Value – A value is the answer of an array. This is usually represented as v(A).

Pilot – In Entry Swap arrays, the pilot is the first entry of the array.

Passengers – In Entry Swap arrays, the passengers are any entries that aren’t the pilot.

Departee – In Entry Swap arrays, the departee is the entry that will be removed from the array or expanded upon.

VIP – The largest passengers in the Entry Swap arrays are considered VIP’s. There can be many VIP’s in the same array, and they are represented as b in the rules. Rule 6 assumes that the b that is moved is the last b in the array.

Seat Swap – In Entry Swap arrays, a seat swap is when the VIP swaps with the departee.

Rules
1 – {n}=n

2 – {n, 0}=The nth Ackermann number

3 – {n, 0, 0}={{{...{{{n, 0}, 0}, 0}, ...0}, 0}, 0} (n bracket pairs)

4 – {n, 0, 0, 0, ...0, 0}={{{...{{{n, 0, 0, 0, ...0}, 0, 0, 0, ...0}, 0, 0, 0, ...0}, ...0, 0, 0, ...0}, 0, 0, 0, ...0}, 0, 0, 0, ...0} (n bracket pairs)

5 – {n, ...b}={{{...{{{n, ...b-1, b-1, ...b-1}, ...b-1, b-1, ...b-1}, ...b-1, b-1, ...b-1}, ... ...b-1, b-1, ...b-1}, ...b-1, b-1, ...b-1} ...b-1, b-1, ...b-1} ({n, ...b-1, b-1, ...n (b-1)’s...b-1) bracket pairs, each with {n, ...b-1, b-1, ...n (b-1)’s...b} (b-1)’s)

6 – {n, ...b, x, ...}={{...{{n, ...x, x, ...x, b, ...}, ...x, x, ...x, b, ...}, ... ...x, x, ...x, b, ...}, x, x, ...x, b, ...}    ({n, ...x, x, ...{n, ...} x’s...x, b, ...} bracket pairs, where b and x are swapped, and in the space where b used to be are {n, ...x, x, ...{n, ...} x’s...x, b, ...} x’s)

Examples
{5, 0}=5^^^^^5

{5, 0, 0, 0}={{{{{5, 0, 0}, 0, 0}, 0, 0}, 0, 0}, 0, 0}={{{{{{{{{5, 0}, 0}, 0}, 0}, 0}, 0, 0}, 0, 0}, 0, 0}, 0, 0}=     {{{{{{{{5^^^^^5, 0}, 0}, 0}, 0}, 0, 0}, 0, 0}, 0, 0}, 0, 0}

{8, 2, 1, 0}={{...{8, 1, 1, 1, 1, 1, 1, 1, 1, 2, 0} bracket pairs...{{8, 1, 1, ...{8, 1, 1, 1, 1, 1, 1, 1, 1, 2, 0} 1’s...1, 2, 0}, 1, 1, ...{8, 1, 1, 1, 1, 1, 1, 1, 1, 2, 0} 1’s...1, 2, 0}, ...1, 1, ...{8, 1, 1, 1, 1, 1, 1, 1, 1, 2, 0} 1’s...1, 2, 0}, 1, 1, ...{8, 1, 1, 1, 1, 1, 1, 1, 1, 2, 0} 1’s...1, 2, 0}

{3, 2, 3, 6, 2}={{...{3, 2, 3, 2, 2, ...{3, 2, 3} 2’s...2, 6} bracket pairs...{{3, 2, 3, 2, 2, ...{3, 2, 3, 2, 2, ...{3, 2, 3} 2’s...2, 6} 2’s...2, 6}, 2, 3, 2, 2, ...{3, 2, 3, 2, 2, ...{3, 2, 3} 2’s...2, 6} 2’s...2, 6}, ...2, 3, 2, 2, ...{3, 2, 3, 2, 2, ...{3, 2, 3} 2’s...2, 6} 2’s...2, 6}, 2, 3, 2, 2, ...{3, 2, 3, 2, 2, ...{3, 2, 3} 2’s...2, 6} 2’s...2, 6}

{10, 10, 10, 1, 10}={{...{10, 10, 10, 1, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9} bracket pairs...{{10, 10, 10, 1, 9, ...{10, 10, 10, 1, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9} 9’s...9}, 10, 10, 1, 9, ...{10, 10, 10, 1, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9} 9’s...9}, ...10, 10, 1, 9, ...{10, 10, 10, 1, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9} 9’s...9}, 10, 10, 1, 9, ...{10, 10, 10, 1, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9} 9’s...9}