Linear array notation (LAN) is the first part of successor array notation.

Definition

Define an array as any (possibly empty) comma-separated list of non-negative integers (entries). Let @ denote 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 basei>.

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. Replace the "0, b" with "a, b - 1" where a is the base and b is the nonzero entry. In simpler terms: a{0,0...,0,0,b,@} = a{0,0...,0,a,b-1,@} if b > 0. The process ends.

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

Note: This notation is similar to "Middle-growing hierarchy" by Ikosarakt1, especially for 1 entry since m(b, a) = a{b} for finite b. I chose to make this notation more like a middle-growing hierarchy extension instead of more like an up-arrow notation extension because it leads to a simpler definition while still having the same growth rate in the end.

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{} = f0(a)
a{1} = f02(a)
a{2} = f04(a)
a{3} = f08(a)
a{4} = f016(a)
a{5} = f032(a)
a{0, 1} ≈ f2(a)
a{1, 1} ≈ f22(a)
a{2, 1} ≈ f24(a)
a{3, 1} ≈ f28(a) a{4, 1} ≈ f216(a)
a{0, 2} ≈ f3(a)
a{1, 2} ≈ f32(a)
a{2, 2} ≈ f34(a)
a{3, 2} ≈ f38(a)
a{4, 2} ≈ f316(a)
a{0, 3} ≈ f4(a)
a{1, 3} ≈ f42(a)
a{0, 0, 1} ≈ fω(a)
a{1, 0, 1} ≈ fω2(a)
a{0, 1, 1} ≈ fω+1(a)
a{1, 1, 1} ≈ fω+12(a)
a{0, 2, 1} ≈ fω+2(a)
a{0, 3, 1} ≈ fω+3(a)
a{0, 4, 1} ≈ fω+4(a)
a{0, 0, 2} ≈ fω2(a)
a{0, 1, 2} ≈ fω2+1(a)
a{0, 0, 3} ≈ fω3(a)
a{0, 0, 4} ≈ fω4(a)
a{0, 0, 0, 1} ≈ fω2(a)
a{1, 0, 0, 1} ≈ fω22(a)
a{0, 1, 0, 1} ≈ fω2+ω(a)
a{0, 0, 1, 1} ≈ fω2+ω+ω(a)
a{0, 0, 0, 2} ≈ fω22(a)
a{0, 0, 0, 3} ≈ fω23(a)
a{0, 0, 0, 4} ≈ fω24(a)
a{0, 0, 0, 0, 1} ≈ fω3(a)
a{0, 0, 0, 0, 2} ≈ fω32(a)
a{0, 0, 0, 0, 0, 1} ≈ fω4(a)