Linear Array Notation is a simple part of my array notation.

Definition

The linear array notation has the following form:

a[c,d,e,f,...,m,n]b where b > 1; c,d,e,f,...,m,n > 1. They can have many number of entries.

The number before array is the base, the number after it is the iterator.

Rules

To solve the expression, we need some following basic rules:

1. Base Rule: a[0]b = a+1, a[1]b = a+b, a[2]b = ab, a[3]b = a^b

2. Tailing Rule: a[#,1]b = a[#]b

3. Prime Rule: a[%]1 = a

4. Recursion Rule: a[c%]b = a[c-1%](a[c%](b-1)), provided c ≥ 0 and b ≥ 1.

Otherwise, you start the process described below from the first number inside the brackets:

1. If the first entries are 1, a[1,1,...,1,1,c,d%]b = a[a,a,...,a,b,c-1,d%]a

2. If the first entries are greater than 1, a[c,d,e,...,n%]b = a[c-1,d,e,...,n%](a[c,d,e,...,n%](b-1))

Explanation

The one-entry array is the same as the hyper-n operators by Robert Munafo.

2-entry arrays

• a[1,2]b is the smallest 2-entry array. It's straightforward! Just replace [1,2] with the basic array using the expression a[b]a. FGH level ω.

• a[2,2]b = a[1,2](a[2,2](b-1)) = a[1,2]a[1,2]...[1,2]a[1,2]a with b copies of a. FGH level ω+1.

• a[3,2]b = a[2,2]a[2,2]...[2,2]a[2,2]a with b copies of a, and so on. FGH level ω+2.

• a[1,3]b = a[b,2]a. FGH level ω2.

• a[2,3]b = a[1,3]a[1,3]...[1,3]a with b copies of a, a[3,3]b = a[2,3]a[2,3]...[2,3]a with b copies of a, and so on. FGH level ω2+1 and ω2+2, respectively.

• a[1,4]b = a[b,3]a, a[1,5]b = a[b,4]a, a[1,6]b = a[b,5]a, and so on. FGH level ω3, ω4, and ω5, respectively.

• a[c,d]b has the FGH level of ω(d-1)+(c-1).

3-entry arrays

• a[1,1,2]b = a[a,b]a. FGH level ω^2.

• a[2,1,2]b = a[1,1,2]a[1,1,2]...[1,1,2]a with b copies of a. FGH level ω^2+1.

• a[3,1,2]b = a[2,1,2]a[2,1,2]...[2,1,2]a with b copies of a. FGH level ω^2+2.

• a[1,2,2]b = a[b,1,2]a. FGH level ω^2+ω.

• a[2,2,2]b = a[1,2,2]a[1,2,2]...[1,2,2]a with b copies of a. FGH level ω^2+ω+1.

• a[1,3,2]b = a[b,2,2]a. FGH level ω^2+ω2.

• a[1,4,2]b = a[b,3,2]a. FGH level ω^2+ω3.

• a[1,1,3]b = a[a,b,2]a. FGH level (ω^2)2.

• a[2,1,3]b = a[1,1,3]a[1,1,3]...[1,1,3]a. FGH level (ω^2)2+1.

• a[1,2,3]b = a[b,1,3]a. FGH level (ω^2)2+ω.

• a[1,3,3]b = a[b,2,3]a. FGH level (ω^2)2+ω2.

• a[1,1,4]b = a[a,b,3]a. FGH level (ω^2)3.

• a[1,1,5]b = a[a,b,4]a. FGH level (ω^2)4.

Higher linear arrays

• a[1,1,1,2]b = a[a,a,b]a. FGH level ω^3.

• a[2,1,1,2]b = a[1,1,1,2]a[1,1,1,2]...[1,1,1,2]a with b copies of a. FGH level ω^3+1.

• a[1,2,1,2]b = a[b,1,1,2]a. FGH level ω^3+ω

• a[1,3,1,2]b = a[b,2,1,2]a. FGH level ω^3+ω2.

• a[1,1,2,2]b = a[a,b,1,2]a. FGH level ω^3+ω^2.

• a[1,2,2,2]b = a[b,1,2,2]a. FGH level ω^3+ω^2+ω.

• a[1,1,3,2]b = a[a,b,2,2]a. FGH level ω^3+(ω^2)2.

• a[1,1,4,2]b = a[a,b,3,2]a. FGH level ω^3+(ω^2)3.

• a[1,1,1,3]b = a[a,a,b,2]a. FGH level (ω^3)2.

• a[1,2,1,3]b = a[b,1,1,3]a. FGH level (ω^3)2+ω.

• a[1,1,2,3]b = a[a,b,1,3]a. FGH level (ω^3)2+ω^2.

• a[1,1,3,3]b = a[a,b,2,3]a. FGH level (ω^3)2+(ω^2)2.

• a[1,1,1,4]b = a[a,a,b,3]a. FGH level (ω^3)3.

• a[1,1,1,5]b = a[a,a,b,4]a. FGH level (ω^3)4.

• a[1,1,1,1,2]b = a[a,a,a,b]a. FGH level ω^4.

• a[1,1,1,2,2]b = a[a,a,b,1,2]a. FGH level ω^4+ω^3.

• a[1,1,1,1,3]b = a[a,a,a,b,2]a. FGH level (ω^4)2.

• a[1,1,1,1,4]b = a[a,a,a,b,3]a. FGH level (ω^4)3.

• a[1,1,1,1,1,2]b = a[a,a,a,a,b]a. FGH level ω^5.

• a[1,1,1,1,1,1,2]b = a[a,a,a,a,a,b]a. FGH level ω^6.

In general, if the array has n 1's, the growth rate is ω^n in the FGH, and the limit of the notation level is ω^ω.