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.'
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.
a[]#b (no matter what does a or b equal to) = 1
1[#]@a = 1
a[#]@1 = a
a[#]@b = {a, b [#] @}
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)
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