Large Expanding Array Notation

Rules:

1. [a, b, 1] = (a^b)*b

2. [a, 1, b, c ... n] = a

3. [a,b,c] = (a^(b^c))*(b*c)

4. [a, b, c ... n] = (a^(b^(c ... n)))*(((n-1 ... c)*b)*a)

5. [a(1)1] = a

6. [a(1)b] = [a, a, a...] (b a's)

7. [a(2)1] = [a(1)a]

8. [a(2)b] = [a(1)a(1)a...]

9. [a(c)b] =  [a(c-1)a(c-1)a(c-1)a...] (b times)

10. [a(a)a] =  [a((1))1]

11. [a((1))b] = [a(a)a(a)...] (b times)

12. [a((c))b] = [a((c-1))a((c-1))a((c-1))a((c-1))a...] (b times)

13. [a@b] = [a((...((a))...))a] (b times)

14. [a@b@c] = [a((...((a))...))a] ([b@c] times)

15. [a@b@c...] = [a((...((a))...))a] ([b@c@d...] times) (The limit)

Examples:

[1, 1] = 1

[5, 5] = (5^5)*5 = 15625

[3, 3, 3] = (3↑↑3)*(3*3)

[4, 4, 4, 4] = (4↑↑4)*(4^3)

[10, 10, 10, 10, 10, 10, 10, 10, 10, 10] = (10^^10)*(10^9)

[10(1)10] = [10, 10, 10, 10, 10, 10, 10, 10, 10, 10]

[3((1))1] = [3(3)3]

[3((1))2]= [3(3)3(3)3]

[3@3] =  [3(((3)))3]

Factoriexponentiated Notation

Rules:

Fe(1, #) = Fe(1)

Fe(#, 1) = Fe(#)

Fe(#[#]1) = Fe(1)

The Meaning:

Pre-brackets

Fe(n) = n^(n!)

Fe(1, #) = Fe(1)

Fe(#, 1) = Fe(#)

Fe(n, 2) = Fe(Fe(n)^(n!)) 

Fe(n, m) = Fe(Fe(n, m-1)^(n!))

Fe(n, m, 2) = Fe(Fe(n, m-1)^(n!))^2

Fe(n, m, o) = Fe(Fe(n, m-1)^(n!))^o

Fe(n, m, o, p) = (Fe(Fe(n, m-1)^(n!)^o)^^p

Fe(n, m, o, p, q) = (Fe(n, m, o, p)^^^q)

First Stage Brackets

Fe(n[1]2) = Fe(n, n, n, n...) (n times)

Fe(n[1]3) = Fe(n[1]2)^(n^(n!))

Fe(n[1]m) = Fe(n[1]m-1)^(n{m-2}(n!))

Fe(n[2]2) = Fe(n[1]n) = Fe(n[1]n-1)^(n{n}(n!))

Fe(n[2]3) = Fe(n[2]2)^(n{n}(n!))

Fe(n[m]o) = Fe(n[m-1]o-1)^(n{n}(n!))

Second Stage Brackets

Fe(n[1,2]1) = Fe(n[n]n)^(n!)

Fe(n[1,2]m) = Fe(n[1,2]m-1)^(((...(n)!)!)...!) (m times)

Fe(n[1,3]1) = Fe(n[1,2]n){n}(n!)

Fe(n[1,3]m) = Fe(n[1,3]m-1){n}(n!)

Fe(n[1,m]o) = Fe(n[1, m]o-1){n}(n!)

Fe(n[2,1]1) = Fe(n[1,n]n){{1}}(n!)

Fe(n[2,1]m) = Fe(n[2,1]m-1){{1}}(n!)

Fe(n[2,m]1) = Fe(n[2,m-1]n){{1}}(n!)

Fe(n[2,m]o) = Fe(n[2,m]o-1){{1}}(n!)

Fe(n[3,1]1) = Fe(n[2,n]n){{{1}}}(n!)

Fe(n[m,1]1) = {Fe(n[m-1,n]n), (n!), 1, m}

Fe(n[m,o]p) = {Fe(n[m,o]p-1), (n!), 1, m}

Third stage and above

Fe(n[1, 1, 2]1) = Fe(n[n,n]n-1) = {Fe(n[n,n]n-1), (n!), 1, n}

Felt very lazy doing this so it will be finished later.