Dimensional array notation (DAN) is the second part of successor array notation.

Definition

Define an array as any (possibly empty) list of non-negative integers (entries) separated by separators. A separator can be any non-negative integer wrapped in round braces, like (3). The comma may be used a shorthand for the separator (0). Let @ represent an arbitrary array. A valid expression is of the form a{@}, where a can be any non-negative integer and is referred to as the "base".

  1. a{} = a + 1
  2. a{@(n)0} = a{@} (remove trailing zeros)
  3. a{b(n)@} = a{b - 1(n)@}{b - 1(n)@} if b > 0

If none of the rules apply, start this process starting from the first entry inside the curly brackets:

  1. If the entry is equal to 0, jump to the next entry.
  2. If the entry is greater than 0, there must be a 0 preceding it.
    1. if there is a comma between the 0 and the nonzero entry, replace the "0,b" with "a,b-1" where a is the base and b is the nonzero entry. The process ends.
    2. if there is a non-comma separator (n) between the 0 and the nonzero entry, replace the "0(n)b" with "a(n - 1)a(n - 1)...a(n-1)a(n)b-1" with b copies of a before the (n), where a is the base and b is the nonzero entry. The process ends.

The application of the rules and/or process should be repeated until the expression is reduced to a number.

Comparison to fast-growing hierarchy

I estimate that the limit of this notation is around ωω in the FGH with the Wainer hierarchy as the system of fundamental sequences. Below I will give the comparison of this notation to the fast-growing hierarchy with the Wainer hierarchy as the system of fundamental sequences:

a{0(1)1} ≈ fωω(a)
a{1(1)1} ≈ fωω2(a)
a{2(1)1} ≈ fωω4(a)
a{3(1)1} ≈ fωω8(a)
a{4(1)1} ≈ fωω16(a)
a{0,1(1)1} ≈ fωω+1(a)
a{1,1(1)1} ≈ fωω+12(a)
a{2,1(1)1} ≈ fωω+14(a)
a{0,2(1)1} ≈ fωω+2(a)
a{0,3(1)1} ≈ fωω+3(a)
a{0,4(1)1} ≈ fωω+4(a)
a{0,0,1(1)1} ≈ fωω+ω(a)
a{0,1,1(1)1} ≈ fωω+ω+1(a)
a{0,0,2(1)1} ≈ fωω+ω2(a)
a{0,0,3(1)1} ≈ fωω+ω3(a)
a{0,0,4(1)1} ≈ fωω+ω4(a)
a{0,0,0,1(1)1} ≈ fωω+ω2(a)
a{0,0,0,2(1)1} ≈ fωω+ω22(a)
a{0,0,0,0,1(1)1} ≈ fωω+ω3(a)
a{0(1)2} ≈ fωω2(a)
a{1(1)2} ≈ fωω22(a)
a{0,1(1)2} ≈ fωω2+1(a)
a{0,0,1(1)2} ≈ fωω2+ω(a)
a{0(1)3} ≈ fωω3(a)
a{0(1)4} ≈ fωω4(a)
a{0(1)0,1} ≈ fωω+1(a)
a{0(1)0,2} ≈ fωω+12(a)
a{0(1)0,0,1} ≈ fωω+2(a)
a{0(1)0(1)1} ≈ fωω2(a)
a{0(1)0(1)0(1)1} ≈ fωω3(a)
a{0(1)0(1)0(1)0(1)1} ≈ fωω4(a)
a{0(2)1} ≈ fωω2(a)
a{1(2)1} ≈ fωω22(a)
a{0,1(2)1} ≈ fωω2+1(a)
a{0,0,1(2)1} ≈ fωω2+ω(a)
a{0(1)1(2)1} ≈ fωω2+ωω(a)
a{0(2)2} ≈ fωω22(a)
a{0(2)3} ≈ fωω23(a)
a{0(2)4} ≈ fωω24(a)
a{0(2)0,1} ≈ fωω2+1(a)
a{0(2)0,0,1} ≈ fωω2+2(a)
a{0(2)0(1)1} ≈ fωω2+ω(a)
a{0(3)1} ≈ fωω3(a)
a{0(3)2} ≈ fωω32(a)
a{0(3)0(2)1} ≈ fωω3+ω2(a)
a{0(4)1} ≈ fωω4(a)