I have a way to make Almighty Array Notation (AAN) stronger.
I know that <n> is a shorthand for !!! … !!! with n !'s.
But that's just secondary extended first-order array notation.
We can have <1 ! 2>, which expands to <1 {1 < … <1 {1 <1> 2} 2> … > 2}2>.
With b – 1 1{1's.
The <<1>> is the first separator of tertiary extended first-order array notation.
And then <<1 ! 2>> expands to <<1 {1 <<1 {1 << … 1{1 <<1>> 2} 2 … >> 2} 2>> 2} 2>> with b – 1 1{1's.
<<<1>>> expands to << 1 << … <<1>> … >>1 >> with b – 1 >>'s.
{1 @ 2} is now << … <<1>> … >> with b – 1 sets of <>.
The @ is a shorthand for <1>2
We can extend it to << … <<1>> … >>2
But how does it work?
<1 ! 2>2 = <1 {1 {1 < … 1 {1 {1 <<1>2 2} 2} 2 … > 2} 2} 2>2
<1 @ 2>2 = <1 {1 < … 1 {1 <1>2 2} 2 … >2 2} 2>2
Now, <<1>>2 = <1 <1 < … <1 <1>2 2> … >2 2> 2>2
<< … <<1>> … >>2 with b – 1 >'s is <<1>2>2.
<< … <<1>>2>> … >>2 with b – 1 >>2's is {1 !3 2} = {1 # 2}
Where !1 = !
And !2 = @
The !3 = #, it’s a shorthand for <1>3.
So we can say the !n is a shorthand for <1>n.
And {1 !n 2} = <<< … <<1>n – 1 > … > n – 1 > n – 1 > n – 1 with b – 1 > n – 1's.
Then we have !A
Where A is an array.
And {1 / 2} = {1 !(1 {1 !(…) 2} 2) 2}
And the / is a shorthand for {X^X}.
{X^n} = !n
{X^X} = {X^1 {X^ …} 2}
{X^^1, 2} = {X^^b}
{X^^X} = {X^^1 {X^^1 {…} 2} 2}
The limit of this is {a*(X, X {a*(X, X {…} 2)} 2)}
We can now let a**(n) = a*(n, n {a*(X, X {…} 2)} 2) where the separator is the limit of the X's.
Now take a*^n(k) = a** … **(k) with n *'s.
a*(n)2 = a*^*^* … ^*^n (n)
The limit of this notation is a*(n)___…
(Referred to the comment by BryantOfOrdinals in this YouTube video [modified])