Hayden's Extended Array Notation

Hayden's Extended Array Notation (HEAN) is a notation based on BAN made by googology wiki user HaydenTheGoogologist2009. It’s an extension of Hayden’s Array Notation. It comes in multiple parts. An example of a valid array is 3[1, 2]23.

Unfortunately, I abandoned this notation on September 15th, 2022 with the statement 'I created this notation when I was too immature.'

Rules

Here, we explain the definitions of first four functions in Hayden's Extended Array Notation, which Hayden specifies as "Rules".

Note that # and @ represents an array.

1. a[]#b (no matter what does a or b equal to) = 1

2. 1[#]@a = 1

3. a[#]@1 = a

4. a[#]@b = {a, b [#] @}

Definition

Here, we explain the definitions of all other functions in Hayden's Extended Array Notation. There are a few things to note:

• All operations are solved from left to right.

• {} indicates BAN (only Nested Hyper-Nested Array Notation is allowed)

a[]cb = 1

a[1]cb = a

a[c]1b = a

a[2]2b = {a, b [2] 2}

a[2]2b / c = {a, b, c [2] 2}

a[2]2b / c / d = {a, b, c, d [2] 2}

a[2]2b / c / d / … x / y / z = {a, b, c, d … x, y, z [2] 2}

a[2]3b = {a, b [2] 3}

a[2]4b = {a, b [2] 4}

a[2]5b = {a, b [2] 5}

a[2]cb = {a, b [2] c}

a[2]1, 2b = {a, b [2] 1, 2}

a[2]2, 2b = {a, b [2] 2, 2}

a[2]3, 2b = {a, b [2] 3, 2}

a[2]1, 3b = {a, b [2] 1, 3}

a[2]2, 3b = {a, b [2] 2, 3}

a[2]3, 3b = {a, b [2] 3, 3}

a[2]1, 4b = {a, b [2] 1, 4}

a[2]1, 5b = {a, b [2] 1, 5}

a[2]1, 1, 2b = {a, b [2] 1, 1, 2}

a[2]2, 1, 2b = {a, b [2] 2, 1, 2}

a[2]1, 2, 2b = {a, b [2] 1, 2, 2}

a[2]1, 1, 3b = {a, b [2] 1, 1, 3}

a[2]1, 1, 1, 2b = {a, b [2] 1, 1, 1, 2}

a[2]1, 1, 1, 1, 2b = {a, b [2] 1, 1, 1, 1, 2}

a[2]c, d, e, … x, y, zb = {a, b [2] c, d, e, … x, y, z}

a[2]1[2]2b = {a, b [2] 1 [2] 2}

a[2]2[2]2b = {a, b [2] 2 [2] 2}

a[2]3[2]2b = {a, b [2] 3 [2] 2}

a[2]1[2]1[2]2b = {a, b [2] 1 [2] 1 [2] 2}

a[2]2[2]1[2]2b = {a, b [2] 2 [2] 1 [2] 2}

a[2]1[2]2[2]2b = {a, b [2] 1 [2] 2 [2] 2}

a[2]1[2]1[2]1[2]2b = {a, b [2] 1 [2] 1 [2] 1 [2] 2}

a[2]1[2]1[2]1[2]1[2]2b = {a, b [2] 1 [2] 1 [2] 1 [2] 1 [2] 2}

a[2]c[2]d[2]e … [2]y[2]z[2]2b = {a, b [2] c [2] d [2] e … [2] y [2] z [2] 2}

a[3]2b = {a, b [3] 2}

a[4]2b = {a, b [4] 2}

a[5]2b = {a, b [5] 2}

a[c]2b = {a, b [c] 2}

a[1, 2]2b = {a, b [1, 2] 2}

a[2, 2]2b = {a, b [2, 2] 2}

a[3, 2]2b = {a, b [3, 2] 2}

a[1, 3]2b = {a, b [1, 3] 2}

a[2, 3]2b = {a, b [2, 3] 2}

a[3, 3]2b = {a, b [3, 3] 2}

a[1, 4]2b = {a, b [1, 4] 2}

a[1, 5]2b = {a, b [1, 5] 2}

a[1, 1, 2]2b = {a, b [1, 1, 2] 2}

a[2, 1, 2]2b = {a, b [2, 1, 2] 2}

a[1, 2, 2]2b = {a, b [1, 2, 2] 2}

a[1, 1, 3]2b = {a, b [1, 1, 3] 2}

a[1, 1, 1, 2]2b = {a, b [1, 1, 1, 2] 2}

a[1, 1, 1, 1, 2]2b = {a, b [1, 1, 1, 1, 2] 2}

a[c, d, e, … x, y, z]2b = {a, b [c, d, e, … x, y, z] 2}

a[1 [2] 2]2b = {a, b [1 [2] 2] 2}

a[2 [2] 2]2b = {a, b [2 [2] 2] 2}

a[1 [2] 3]2b = {a, b [1 [2] 3] 2}

a[1 [3] 2]2b = {a, b [1 [3] 2] 2}

a[1 [1, 2] 3]2b = {a, b [1 [1, 2] 3] 2}

a[1 [1, 1, 2] 3]2b = {a, b [1 [1, 1, 2] 3] 2}

a[1 [1 [2] 1] 2]2b = {a, b [1 [1 [2] 2] 2] 2}

a[1 [1 [1 [2] 2] 2] 2]2b = {a, b [1 [1 [1 [2] 2] 2] 2] 2}

a[c [d [e … [x [y [z] y] x] … e] d] c]2b = {a, b [c [d [e … [x [y [z] y] x] … e] d] c] 2}

a[c [d [e … [y [z [α, β, γ, … χ, ψ, ω] z] y] … e] d] c]2b = {a, b [c [d [e … [y [z [α, β, γ, … χ, ψ, ω] z] y] … e] d] c] 2} (all Greek alphabets represents an integer just like the Latin alphabets)

a[1 \ 2]2b = {a, b [1 \ 2] 2}

a[2 \ 2]2b = {a, b [2 \ 2] 2}

a[1 [1, 2 \ 2] 2 \ 2]2b = {a, b [1, [1, 2 \ 2] 2 \ 2] 2}

a[1 [1 [1 \ 2] 1, 2 \ 2] 2 \ 2]2b = {a, b [1 [1 [1 \ 2] 1, 2 \ 2] 2 \ 2] 2}

a[3 \ 2]2b = {a, b [3 \ 2] 2}

a[1, 2 \ 2]2b = {a, b [1, 2 \ 2] 2}

a[2, 2 \ 2]2b = {a, b [2, 2 \ 2] 2}

a[3, 2 \ 2]2b = {a, b [3, 2 \ 2] 2}

a[1, 3 \ 2]2b = {a, b [1, 3 \ 2] 2}

a[2, 3 \ 2]2b = {a, b [2, 3 \ 2] 2}

a[3, 3 \ 2]2b = {a, b [3, 3 \ 2] 2}

a[1, 4 \ 2]2b = {a, b [1, 4 \ 2] 2}

a[1, 5 \ 2]2b = {a, b [1, 5 \ 2] 2}

a[1, 1, 2 \ 2]2b = {a, b [1, 1, 2 \ 2] 2}

a[2, 1, 2 \ 2]2b = {a, b [2, 1, 2 \ 2] 2}

a[1, 2, 2 \ 2]2b = {a, b [1, 2, 2 \ 2] 2}

a[1, 1, 3 \ 2]2b = {a, b [1, 1, 3 \ 2] 2}

a[1, 1, 1, 2 \ 2]2b = {a, b [1, 1, 1, 2 \ 2] 2}

a[1, 1, 1, 1, 2 \ 2]2b = {a, b [1, 1, 1, 1, 2 \ 2] 2}

a[1 [2] 2 \ 2]2b = {a, b [1 [2] 2 \ 2] 2}

a[1 [2] 1, 2 \ 2]2b = {a, b [1 [2] 1, 1 \ 2] 2}

a[1 [2] 1, 1, 1 \ 2]2b = {a, b [1 [2] 1, 1, 1 \ 2] 2}

a[1 [2]1[2] 1 \ 2]2b = {a, b [1 [2] 1 [2] 1 \ 2] 2}

a[1 \ 3]2b = {a, b [1 \ 3] 2}

a[2 \ 3]2b = {a, b [2 \ 3] 2}

a[3 \ 3]2b = {a, b [3 \ 3] 2}

a[1 \ 4]2b = {a, b [1 \ 4] 2}

a[1 \ 5]2b = {a, b [1 \ 5] 2}

a[1 \ 1, 2]2b = {a, b [1 \ 1, 2] 2}

a[2 \ 1, 2]2b = {a, b [2 \ 1, 2] 2}

a[3 \ 1, 2]2b = {a, b [3 \ 1, 2] 2}

a[1 [1 \ 1, 2] 2 \ 1, 2]2b = {a, b [1 [1 \ 1, 2] 2 \ 1, 2] 2}

a[1 \ 2, 2]2b = {a, b [1 \ 2, 2] 2}

a[2 \ 2, 2]2b = {a, b [2 \ 2, 2] 2}

a[3 \ 2, 2]2b = {a, b [3 \ 2, 2] 2}

a[1 \ 3, 2]2b = {a, b [1 \ 3, 2] 2}

a[2 \ 3, 2]2b = {a, b [2 \ 3, 2] 2}

a[3 \ 3, 2]2b = {a, b [3 \ 3, 2] 2}

a[1 \ 1, 3]2b = {a, b [1 \ 1, 3] 2}

a[2 \ 1, 3]2b = {a, b [2 \ 1, 3] 2}

a[3 \ 1, 3]2b = {a, b [3 \ 1, 3] 2}

a[1 \ 1, 3]2b = {a, b [1 \ 2, 3] 2}

a[2 \ 2, 3]2b = {a, b [2 \ 2, 3] 2}

a[3 \ 2, 3]2b = {a, b [3 \ 2, 3] 2}

a[1 \ 1, 4]2b = {a, b [1 \ 1, 4] 2}

a[1 \ 1, 5]2b = {a, b [1 \ 1, 5] 2}

a[1 \ 1, 1, 2]2b = {a, b [1 \ 1, 1, 2] 2}

a[1 \ 1, 2, 2]2b = {a, b [1 \ 1, 2, 2] 2}

a[1 \ 1, 1, 3]2b = {a, b [1 \ 1, 1, 3] 2}

a[1 \ 1, 1, 1, 2]2b = {a, b [1 \ 1, 1, 1, 2] 2}

a[1 \ 1, 1, 1, 1, 2]2b = {a, b [1 \ 1, 1, 1, 1, 2] 2}

a[1 \ 1 [2] 2]2b = {a, b [1 [2] 2 \ 2] 2}

a[1 \ 1 [2] 1, 2]2b = {a, b [1 \ 1 [2] 1, 2] 2}

a[1 \ 1 [2] 1, 1, 2]2b = {a, b [1 \ 1 [2] 1, 1, 2] 2}

a[1 \ 1 [2]1[2] 2]2b = {a, b [1 \ 1 [2] 1 [2] 2] 2}

a[1 \ 1 \ 2]2b = {a, b [1 \ 1 \ 2] 2}

a[2 \ 1 \ 2]2b = {a, b [2 \ 1 \ 2] 2}

a[1 \ 2 \ 2]2b = {a, b [1 \ 2 \ 2] 2}

a[1 \ 1 \ 3]2b = {a, b [1 \ 1 \ 3] 2}

a[1 \ 1 \ 1 \ 2]2b = {a, b [1 \ 1 \ 1 \ 2] 2}

a[1 \ 1 \ 1 \ 1 \ 2]2b = {a, b [1 \ 1 \ 1 \ 1 \ 2] 2}

a[d \ e \ f \ … x \ y \ z]cb = {a, b [d \ e \ f \ … x \ y \ z] c}

a[d \ e \ f \ … y \ z \ α, β, Γ, … χ, ψ, ω]2b = {a, b [d \ e \ f \ … y \ z \ α, β, Γ, … χ, ψ, ω] 2} (all Greek alphabets represents an integer just like the Latin alphabets)

a[1 [2 ¬ 2] 2]2b = {a, b [1 [2 ¬ 2] 2] 2}

a[1 [1 ¬ 3] 2]2b = {a, b [1 [1 ¬ 3] 2] 2}

a[1 [2 ¬ 3] 1 \ 2]2b = {a, b [1 [2 ¬ 3] 1 \ 2] 2}

a[1 [2 ¬ 3] 2]2b = {a, b [1 [2 ¬ 3] 2] 2}

a[1 [1 ¬ 4] 2]2b = {a, b [1 [1 ¬ 4] 2] 2}

a[1 [1 ¬ 5] 2]2b = {a, b [1 [1 ¬ 5] 2] 2}

a[c [d ¬ e] f]2b = {a, b [c [d ¬ e] f] b}

a[1 [1 [2 \3 2] 2] 2]2b = {a, b [1 [1 [1 \3 2] 2] 2] 2}

a[1 [1 [1 [2 \4 2] 2] 2] 2]2b = {a, b [1 [1 [1 [1 \4 2] 2] 2] 2] 2}

a[1 [1 [1 [1 [2 \5 2] 2] 2] 2] 2]2b = {a, b [1 [1 [1 [1 [1 \5 2] 2] 2] 2] 2] 2}

a[1 [1 [1 … [1 [1 [2 \c 2] 2] 2] … 2] 2] 2]2b (c 1’s on the left, c 2’s on the right) = {a, b [1 [1 [1 … [1 [1 [1 \c 2] 2] 2] … 2] 2] 2] 2} (c 1’s on the left, c 2’s on the right) 

Examples

100[1 \ 2]249 = {100, 49 [1 \ 2] 2} ≈ tethrathoth

100[1 [1, 2 \ 2] 2 \ 2]2100 = {100, 100 [1 [1, 2 \ 2] 2 \ 2] 2} ≈ monster-giant

100[1 \ 1, 2]2100 = {100, 100 [1 \ 1, 2] 2} ≈ tethrathoth ba’al

100[1 \ 1 \ 2]249 = {100, 49 [1 \ 1 \ 2] 2} ≈ tethracross

100[1 \ 1 \ 1 \ 2]249 = {100, 49 [1 \ 1 \ 1 \ 2] 2} ≈ tethracubor

100[1 \ 1 \ 1 \ 1 \ 2]249 = {100, 49 [1 \ 1 \ 1 \ 1 \ 2] 2} ≈ tethrateron

100[1 \ 1 \ 1 \ 1 \ 1 \ 2]249 = {100, 49 [1 \ 1 \ 1 \ 1 \ 1 \ 2] 2} ≈ tethrapeton

100[1 [2 ¬ 2] 2]27 = {100, 7 [1 [2 ¬ 2] 2] 2} ≈ tethrahexon

100[1 [2 ¬ 2] 2]28 = {100, 8 [1 [2 ¬ 2] 2] 2} ≈ tethrahepton

100[1 [2 ¬ 2] 2]29 = {100, 9 [1 [2 ¬ 2] 2] 2} ≈ tethra-ogdon

100[1 [2 ¬ 2] 2]210 = {100, 10 [1 [2 ¬ 2] 2] 2} ≈ tethrennon

100[1 [2 ¬ 2] 2]211 = {100, 11 [1 [2 ¬ 2] 2] 2} ≈ tethradekon

100[1 [2 ¬ 2] 2]2101 = {100, 101 [1 [2 ¬ 2] 2] 2} ≈ tethratope

100[1 [1 ¬ 3] 2]2100 = {100, 100 [1 [1 ¬ 3] 2] 2} ≈ pentacthulhum

100[1 [1 ¬ 3] 1 \ 2]2100 = {100, 100 [1 [1 ¬ 3] 1 \ 2] 2} ≈ pentacthulcross

100[1 [1 ¬ 3] 1 \ 1 \ 2]2100 = {100, 100 [1 [1 ¬ 3] 1 \ 1 \ 2] 2} ≈ pentacthulcubor

100[1 [1 ¬ 3] 1 \ 1 \ 1 \ 2]2100 = {100, 100 [1 [1 ¬ 3] 1 \ 1 \ 1 \ 2] 2} ≈ pentacthulteron

100[1 [1 ¬ 3] 1 \ 1 \ 1 \ 1 \ 2]2100 = {100, 100 [1 [1 ¬ 3] 1 \ 1 \ 1 \ 1 \ 2] 2} ≈ pentacthulteron

100[1 [1 ¬ 3] 1 \ 1 \ 1 \ 1 \ 1 \ 2]2100 = {100, 100 [1 [1 ¬ 3] 1 \ 1 \ 1 \ 1 \ 1 \ 2] 2} ≈ pentacthulpeton

100[1 [1 ¬ 3] 1 [2 ¬ 2] 2]26 = {100, 6 [1 [1 ¬ 3] 1 [2 ¬ 2] 2] 2} ≈ pentacthulhexon

100[1 [1 ¬ 3] 1 [2 ¬ 2] 2]27 = {100, 7 [1 [1 ¬ 3] 1 [2 ¬ 2] 2] 2} ≈ pentacthulhepton

100[1 [1 ¬ 3] 1 [2 ¬ 2] 2]28 = {100, 8 [1 [1 ¬ 3] 1 [2 ¬ 2] 2] 2} ≈ pentacthul-ogdon

100[1 [1 ¬ 3] 1 [2 ¬ 2] 2]29 = {100, 9 [1 [1 ¬ 3] 1 [2 ¬ 2] 2] 2} ≈ pentacthulennon

100[1 [1 ¬ 3] 1 [2 ¬ 2] 2]210 = {100, 10 [1 [1 ¬ 3] 1 [2 ¬ 2] 2] 2} ≈ pentacthuldekon

100[1 [1 ¬ 3] 1 [2 ¬ 2] 2]2100 = {100, 100 [1 [1 ¬ 3] 1 [2 ¬ 2] 2] 2} ≈ pentacthultope

100[1 [1 ¬ 3] 1 [1 ¬ 3] 2]2100 = {100, 100 [1 [1 ¬ 3] 1 [1 ¬ 3] 2] 2} ≈ hexacthulhum

100[1 [1 ¬ 3] 1 [1 ¬ 3] 1 [1 ¬ 3] 2]2100 = {100, 100 [1 [1 ¬ 3] 1 [1 ¬ 3] 1 [1 ¬ 3] 2] 2} ≈ heptacthulhum

100[1 [1 ¬ 3] 1 [1 ¬ 3] 1 [1 ¬ 3] 1 [1 ¬ 3] 2]2100 = {100, 100 [1 [1 ¬ 3] 1 [1 ¬ 3] 1 [1 ¬ 3] 1 [1 ¬ 3] 2] 2} ≈ ogdacthulhum

100[1 [2 ¬ 3] 2]26 = {100, 6 [1 [2 ¬ 3] 2] 2} ≈ ennacthulhum

100[1 [2 ¬ 3] 2]27 = {100, 7 [1 [2 ¬ 3] 2] 2} ≈ dekacthulhum

100[1 [1 ¬ 4] 2]299 = {100, 99 [1 [1 ¬ 4] 2] 2} ≈ blasphemorgulus