Nested array notation (NAN) is the third part of successor array notation.

Definition

The nested successor array notation is a notation which allows reducing an expression made up of several numbers into a single extremely large number, for the purpose of googology. Define an array as any (possibly empty) list of non-negative integers (entries) separated by separators. A separator can be any array wrapped in round braces, like (1,2,3). The comma may be used a shorthand for the separator () or (0). Let @ denote an arbitrary array and let $ denote an arbitrary separator. A valid expression is of the form a{@}, where a can be any non-negative integer and is referred to as the "base". In order to evaluate an expression into a number, the following rules should be used:

1. a{} = a + 1
2. a{@$0} = a{@} (remove trailing zeros)
3. a{b$@} = a{b - 1$@}{b - 1$@} 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$@) where n > 0 between the 0 and the nonzero entry, replace the "0(n$@)b" with "a(n - 1$@)a(n - 1$@)...a(n - 1$@)a(n$@)b-1" with b copies of a before the (n$@), where a is the base and b is the nonzero entry. The process ends.
   3. If there is a non-comma separator (0$@) between the 0 and the nonzero entry, simplify the separator itself, removing trailing zeros (analogous to rule 2 of the notation) and/or applying this process on the numbers inside the separator until the separator cannot be simplified anymore. 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 ε₀ 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(0,1)1} ≈ fωωω(a)
a{1(0,1)1} ≈ fωωω2(a)
a{2(0,1)1} ≈ fωωω4(a)
a{3(0,1)1} ≈ fωωω8(a)
a{4(0,1)1} ≈ fωωω16(a)
a{0,1(0,1)1} ≈ fωωω+1(a)
a{0,0,1(0,1)1} ≈ fωωω+ω(a)
a{0(1)1(0,1)1} ≈ fωωω+ωω(a)
a{0(2)1(0,1)1} ≈ fωωω+ωω2(a)
a{0(0,1)2} ≈ fωωω2(a)
a{0(0,1)3} ≈ fωωω3(a)
a{0(0,1)4} ≈ fωωω4(a)
a{0(0,1)0,1} ≈ fωωω+1(a)
a{0(0,1)0,0,1} ≈ fωωω+2(a)
a{0(0,1)0(1)1} ≈ fωωω+ω(a)
a{0(0,1)0(2)1} ≈ fωωω+ω2(a)
a{0(0,1)0(0,1)1} ≈ fωωω2(a)
a{0(0,1)0(0,1)2} ≈ fωωω22(a)
a{0(0,1)0(0,1)0(0,1)1} ≈ fωωω3(a)
a{0(1,1)1} ≈ fωωω+1(a)
a{0(1,1)2} ≈ fωωω+12(a)
a{0(2,1)1} ≈ fωωω+2(a)
a{0(3,1)1} ≈ fωωω+3(a)
a{0(4,1)1} ≈ fωωω+4(a)
a{0(0,2)1} ≈ fωωω2(a)
a{0(1,2)1} ≈ fωωω2+1(a)
a{0(2,2)1} ≈ fωωω2+2(a)
a{0(3,2)1} ≈ fωωω2+3(a)
a{0(4,2)1} ≈ fωωω2+4(a)
a{0(0,3)1} ≈ fωωω3(a)
a{0(1,3)1} ≈ fωωω3+1(a)
a{0(0,4)1} ≈ fωωω4(a)
a{0(0,0,1)1} ≈ fωωω2(a)
a{0(1,0,1)1} ≈ fωωω2+1(a)
a{0(0,1,1)1} ≈ fωωω2+ω(a)
a{0(0,0,2)1} ≈ fωωω22(a)
a{0(0,0,3)1} ≈ fωωω23(a)
a{0(0,0,4)1} ≈ fωωω24(a)
a{0(0,0,0,1)1} ≈ fωωω3(a)
a{0(0,0,0,0,1)1} ≈ fωωω4(a)
a{0(0(1)1)1} ≈ fωωωω(a)
a{0(1(1)1)1} ≈ fωωωω+1(a)
a{0(2(1)1)1} ≈ fωωωω+2(a)
a{0(3(1)1)1} ≈ fωωωω+3(a)
a{0(4(1)1)1} ≈ fωωωω+4(a)
a{0(0,1(1)1)1} ≈ fωωωω+ω(a)
a{0(0,0,1(1)1)1} ≈ fωωωω+ω2(a)
a{0(0(1)2)1} ≈ fωωωω2(a)
a{0(0(1)3)1} ≈ fωωωω3(a)
a{0(0(1)4)1} ≈ fωωωω4(a)
a{0(0(1)0,1)1} ≈ fωωωω+1(a)
a{0(0(1)0,0,1)1} ≈ fωωωω+2(a)
a{0(0(1)0(1)1)1} ≈ fωωωω2(a)
a{0(0(1)0(1)0(1)1)1} ≈ fωωωω3(a)
a{0(0(2)1)1} ≈ fωωωω2(a)
a{0(0(3)1)1} ≈ fωωωω3(a)
a{0(0(0,1)1)1} ≈ fωωωωω(a)
a{0(0(0(1)1)1)1} ≈ fωωωωωω(a)