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Expanding array notation (EAN) is the third part of my array notation. It may go too far, and be too complex, so it’ll be separated into two small parts. First I introduce “primitive expansion” to you, then the EAN comes.

Primitive expansion

Symbol . is called dot, with ASCII = 46

Now let the dot be a separator (add this to the definition of separators). It works in such a way: when we meet a non-1 entry immediately after a dot, first pull the dot out like the case B2 in exAN, then change the former dot into {1{1…{1{1,2}2}…2}2} with b-1 1’s, where b is the iterator.

So for example,

s(a,b.2) = s(a,b{1{1…{1{1,2}2}…2}2}2.1) = s(a,b{1{1…{1{1,2}2}…2}2}2) with b 2’s
s(a,b. c #) = s(a,b{1{1…{1{1,2}2}…2}2}2. c-1 #) with b 2’s
s(a,b.1.2) = s(a,b.1{1{1…{1{1,2}2}…2}2}2) with b 2’s

Now a new type of separators come. The first one is {1^.} where the dot is at the left-superscript position of the rbrace, and the dot is the shorthand for it.

Then {2^.}, {3^.}, {1,2^.}, {1{1,2}2^.}, {1{1^.}2^.}, {2{1^.}2^.}, {1{1,2}2{1^.}2^.}, {1{1^.}3^.}, {1{1^.}1{1^.}2^.}, {1{2^.}2^.}, {1{3^.}2^.}, {1{1,2^.}2^.}, {1{1{1,2^.}2^.}2^.}, etc. They can be solved using exAN process and the “dot rule”.

{1^.2} works as follows: when we meet a non-1 entry immediately after a {1^.2}, first pull it out like the case B2 in exAN, then change the former {1^.2} into {1{1…{1{1,2^.}2^.}…2^.}2^.} with b-1 1’s, where b is the iterator. Then we have {2^.2}, {1,2^.2}, {1{1^.}2^.2}, {1{2^.}2^.2}, {1{1,2^.}2^.2}, {1{1{1^.}2^.}2^.2}, {1{1{1{1^.}2^.}2^.}2^.2}, {1{1^.2}2^.2}, {1{1{1^.}2^.}2{1^.2}2^.2}, {1{1^.2}3^.2}, {1{1^.2}1{1^.2}2^.2}, {1{2^.2}2^.2}, {1{1,2^.2}2^.2}, {1{1{1^.}2^.2}2^.2}, {1{1{1^.2}2^.2}2^.2}, {1{1{1{1^.2}2^.2}2^.2}2^.2}, etc.

Then {1^.3}, {1^.4}, and so on. They works as follows: when we meet a non-1 entry immediately after a {1^.k}, first pull it out like the case B2 in exAN, then change the former {1^.k} into {1{1…{1{1,2^.k-1}2^.k-1}…2^.k-1}2^.k-1} with b-1 1’s, where b is the iterator. And other separators follow the exAN rules.

So far, we have such a process for the notation. The non-process rules remain the same as exAN.

Process

(Case B1, B2 and B4 are terminal but case A and B3 are not.) First start from the 3rd entry.

Case A: If the entry is 1, then you jump to the next entry.
Case B: If the entry n is not 1, look to your left:
- Case B1: If the comma is immediately before you, then
  1. Change the “1,n” into “b,n-1” where n is this non-1 entry and the b is the iterator.
  2. Change all the entries at base layer before them into the base.
  3. The process ends.
- Case B2: If {1^.k} is immediately before you (see {A^.} as {A^.1}), then
  1. Change the “{1^.k} n” into “S_b 2 {1^.k} n-1″, where b is the iterator, S₁ is comma, and S_i+1 = {1 S_i 2^.k-1} (k > 1) or S_i+1 = {1 S_i 2} (k = 1).
  2. The process ends.
- Case B3: If a separator A that is neither comma nor {1^.k} is immediately before you, then
  1. Change the “A n” into “A 2 A n-1”.
  2. Jump to the first entry of the former A.
- Case B4: If an lbrace is immediately before you, then
  1. Change separator {n #} into string S_b, where b is the iterator, S₁ = “{n-1 #}” and S_i+1 = “S_i 1 {n-1 #}”.
  2. The process ends.

Explanation

By the new case B2, a {1^.} expands into nests of {1 something 2}′s, and a {1^.k} expands into nests of {1 something 2^.k-1}′s. That’s why I call it expansion.

Comparison

For further comparisons between my array notation and FGH, we need to define fundamental sequences for φ function. ${\varphi(0,\beta)=\omega^\beta}$ , so its fundamental sequences are already defined.

${\varphi(\alpha+1,0)[0]=0}$
${\varphi(\alpha+1,0)[n+1]=\varphi(\alpha,\varphi(\alpha+1,0)[n])}$
${\varphi(\alpha+1,\beta+1)[0]=\varphi(\alpha+1,\beta)+1}$
${\varphi(\alpha+1,\beta+1)[n+1]=\varphi(\alpha,\varphi(\alpha+1,\beta+1)[n])}$
${\varphi(\alpha,\beta)[n]=\varphi(\alpha,\beta[n])}$ for limit β
${\varphi(\alpha,0)[n]=\varphi(\alpha[n],0)}$ for limit α
${\varphi(\alpha,\beta+1)[n]=\varphi(\alpha[n],\varphi(\alpha,\beta)+1)}$ for limit α

And note that ${\varepsilon_\alpha=\varphi(1,\alpha)}$ is a part of φ function.

Something notable. Above {2}, for all separator A, the s(n,n A 2) always has growth rate ${\omega^{\omega^\alpha}}$ for some α. We say that A has recursion level α. And s(n,n A k) has growth rate ${\omega^{\omega^\alpha}\times(k-1)}$ .

In s(n,n something A something A … A something A something ), if A has the highest recursion level among all separators at layer 1 of the array, and A has recursion level α, and there’re k A’s at layer 1, then this array has growth rate ${\omega^{\omega^\alpha k}\beta+\gamma}$ , where s(n,n something ) (with the last “something”) has growth rate β (above s(n,n,1,2)), and s(n,n something A something A … A something ) (excluding the last “something” and the last A) has growth rate γ (above s(n,n,1,2)).

And from now on, I use recursion level in place of growth rate (use separator A in place of s(n,n A 2)) in comparisons.

{1^.} has recursion level ${\varepsilon_0}$
{2^.} has recursion level ${\varepsilon_0+1}$
{3^.} has recursion level ${\varepsilon_0+2}$
{1,2^.} has recursion level ${\varepsilon_0+\omega}$
{2,2^.} has recursion level ${\varepsilon_0+\omega+1}$
{1,3^.} has recursion level ${\varepsilon_0+\omega2}$
{1,1,2^.} has recursion level ${\varepsilon_0+\omega^2}$
{2,1,2^.} has recursion level ${\varepsilon_0+\omega^2+1}$
{1,2,2^.} has recursion level ${\varepsilon_0+\omega^2+\omega}$
{1,1,3^.} has recursion level ${\varepsilon_0+\omega^22}$
{1,1,1,2^.} has recursion level ${\varepsilon_0+\omega^3}$
{1{2}2^.} has recursion level ${\varepsilon_0+\omega^\omega}$
{2{2}2^.} has recursion level ${\varepsilon_0+\omega^\omega+1}$
{1,2{2}2^.} has recursion level ${\varepsilon_0+\omega^\omega+\omega}$
{1{2}3^.} has recursion level ${\varepsilon_0+\omega^\omega2}$
{1{2}1,2^.} has recursion level ${\varepsilon_0+\omega^{\omega+1}}$
{2{2}1,2^.} has recursion level ${\varepsilon_0+\omega^{\omega+1}+1}$
{1{2}2,2^.} has recursion level ${\varepsilon_0+\omega^{\omega+1}+\omega^\omega}$
{1{2}1,3^.} has recursion level ${\varepsilon_0+\omega^{\omega+1}2}$
{1{2}1,1,2^.} has recursion level ${\varepsilon_0+\omega^{\omega+2}}$
{1{2}1{2}2^.} has recursion level ${\varepsilon_0+\omega^{\omega2}}$
{1{2}1{2}1{2}2^.} has recursion level ${\varepsilon_0+\omega^{\omega3}}$
{1{3}2^.} has recursion level ${\varepsilon_0+\omega^{\omega^2}}$
{1{4}2^.} has recursion level ${\varepsilon_0+\omega^{\omega^3}}$
{1{1,2}2^.} has recursion level ${\varepsilon_0+\omega^{\omega^\omega}}$
{1{2,2}2^.} has recursion level ${\varepsilon_0+\omega^{\omega^{\omega+1}}}$
{1{1,3}2^.} has recursion level ${\varepsilon_0+\omega^{\omega^{\omega2}}}$
{1{1,1,2}2^.} has recursion level ${\varepsilon_0+\omega^{\omega^{\omega^2}}}$
{1{1,1,1,2}2^.} has recursion level ${\varepsilon_0+\omega^{\omega^{\omega^3}}}$
{1{1{2}2}2^.} has recursion level ${\varepsilon_0+\omega^{\omega^{\omega^\omega}}}$
{1{1{2}3}2^.} has recursion level ${\varepsilon_0+\omega^{\omega^{\omega^\omega2}}}$
{1{1{2}1,2}2^.} has recursion level ${\varepsilon_0+\omega^{\omega^{\omega^{\omega+1}}}}$
{1{1{2}1{2}2}2^.} has recursion level ${\varepsilon_0+\omega^{\omega^{\omega^{\omega2}}}}$
{1{1{3}2}2^.} has recursion level ${\varepsilon_0+\omega^{\omega^{\omega^{\omega^2}}}}$
{1{1{1,2}2}2^.} has recursion level ${\varepsilon_0+\omega^{\omega^{\omega^{\omega^\omega}}}}$
{1{1{1{2}2}2}2^.} has recursion level ${\varepsilon_0+\omega^{\omega^{\omega^{\omega^{\omega^\omega}}}}}$
{1{1{1{1,2}2}2}2^.} has recursion level ${\varepsilon_0+\omega^{\omega^{\omega^{\omega^{\omega^{\omega^\omega}}}}}}$
{1.2^.} has recursion level ${\varepsilon_02}$
{2.2^.} has recursion level ${\varepsilon_02+1}$
{1,2.2^.} has recursion level ${\varepsilon_02+\omega}$
{1{2}2.2^.} has recursion level ${\varepsilon_02+\omega^\omega}$
{1{1,2}2.2^.} has recursion level ${\varepsilon_02+\omega^{\omega^\omega}}$
{1{1{2}2}2.2^.} has recursion level ${\varepsilon_02+\omega^{\omega^{\omega^\omega}}}$
{1{1{1,2}2}2.2^.} has recursion level ${\varepsilon_02+\omega^{\omega^{\omega^{\omega^\omega}}}}$
{1.3^.} has recursion level ${\varepsilon_03}$
{1.4^.} has recursion level ${\varepsilon_04}$
{1.1,2^.} has recursion level ${\varepsilon_0\omega=\omega^{\varepsilon_0+1}}$
{1.2,2^.} has recursion level ${\omega^{\varepsilon_0+1}+\varepsilon_0}$
{1.1,3^.} has recursion level ${\omega^{\varepsilon_0+1}2}$
{1.1,1,2^.} has recursion level ${\omega^{\varepsilon_0+2}}$
{1.1{2}2^.} has recursion level ${\omega^{\varepsilon_0+\omega}}$
{1.1{2}3^.} has recursion level ${\omega^{\varepsilon_0+\omega}2}$
{1.1{2}1,2^.} has recursion level ${\omega^{\varepsilon_0+\omega+1}}$
{1.1{2}1{2}2^.} has recursion level ${\omega^{\varepsilon_0+\omega2}}$
{1.1{3}2^.} has recursion level ${\omega^{\varepsilon_0+\omega^2}}$
{1.1{1,2}2^.} has recursion level ${\omega^{\varepsilon_0+\omega^\omega}}$
{1.1{1{2}2}2^.} has recursion level ${\omega^{\varepsilon_0+\omega^{\omega^\omega}}}$
{1.1{1{1,2}2}2^.} has recursion level ${\omega^{\varepsilon_0+\omega^{\omega^{\omega^\omega}}}}$
{1.1.2^.} has recursion level ${\omega^{\varepsilon_02}}$
{1.1.3^.} has recursion level ${\omega^{\varepsilon_02}2}$
{1.1.1,2^.} has recursion level ${\omega^{\varepsilon_02+1}}$
{1.1.1{2}2^.} has recursion level ${\omega^{\varepsilon_02+\omega}}$
{1.1.1{1,2}2^.} has recursion level ${\omega^{\varepsilon_02+\omega^\omega}}$
{1.1.1.2^.} has recursion level ${\omega^{\varepsilon_03}}$
{1{2^.}2^.} has recursion level ${\omega^{\varepsilon_0\omega}=\omega^{\omega^{\varepsilon_0+1}}}$
{1.2{2^.}2^.} has recursion level ${\omega^{\omega^{\varepsilon_0+1}}+\varepsilon_0}$
{1.3{2^.}2^.} has recursion level ${\omega^{\omega^{\varepsilon_0+1}}+\varepsilon_02}$
{1.1,2{2^.}2^.} has recursion level ${\omega^{\omega^{\varepsilon_0+1}}+\omega^{\varepsilon_0+1}}$
{1.1.2{2^.}2^.} has recursion level ${\omega^{\omega^{\varepsilon_0+1}}+\omega^{\varepsilon_02}}$
{1{2^.}3^.} has recursion level ${\omega^{\omega^{\varepsilon_0+1}}2}$
{1{2^.}1,2^.} has recursion level ${\omega^{\omega^{\varepsilon_0+1}+1}}$
{1{2^.}1.2^.} has recursion level ${\omega^{\omega^{\varepsilon_0+1}+\varepsilon_0}}$
{1{2^.}1.3^.} has recursion level ${\omega^{\omega^{\varepsilon_0+1}+\varepsilon_0}2}$
{1{2^.}1.1,2^.} has recursion level ${\omega^{\omega^{\varepsilon_0+1}+\varepsilon_0+1}}$
{1{2^.}1.1.2^.} has recursion level ${\omega^{\omega^{\varepsilon_0+1}+\varepsilon_02}}$
{1{2^.}1{2^.}2^.} has recursion level ${\omega^{\omega^{\varepsilon_0+1}2}}$
{1{3^.}2^.} has recursion level ${\omega^{\omega^{\varepsilon_0+2}}}$
{1{4^.}2^.} has recursion level ${\omega^{\omega^{\varepsilon_0+3}}}$
{1{1,2^.}2^.} has recursion level ${\omega^{\omega^{\varepsilon_0+\omega}}}$
{1{1{2}2^.}2^.} has recursion level ${\omega^{\omega^{\varepsilon_0+\omega^\omega}}}$
{1{1{1,2}2^.}2^.} has recursion level ${\omega^{\omega^{\varepsilon_0+\omega^{\omega^\omega}}}}$
{1{1.2^.}2^.} has recursion level ${\omega^{\omega^{\varepsilon_02}}}$
{2{1.2^.}2^.} has recursion level ${\omega^{\omega^{\varepsilon_02}}+1}$
{1.2{1.2^.}2^.} has recursion level ${\omega^{\omega^{\varepsilon_02}}+\varepsilon_0}$
{1.1.2{1.2^.}2^.} has recursion level ${\omega^{\omega^{\varepsilon_02}}+\omega^{\varepsilon_02}}$
{1{2^.}2{1.2^.}2^.} has recursion level ${\omega^{\omega^{\varepsilon_02}}+\omega^{\omega^{\varepsilon_0+1}}}$
{1{1.2^.}3^.} has recursion level ${\omega^{\omega^{\varepsilon_02}}2}$
{1{1.2^.}1,2^.} has recursion level ${\omega^{\omega^{\varepsilon_02}+1}}$
{1{1.2^.}1.2^.} has recursion level ${\omega^{\omega^{\varepsilon_02}+\varepsilon_0}}$
{1{1.2^.}1{1.2^.}2^.} has recursion level ${\omega^{\omega^{\varepsilon_02}2}}$
{1{2.2^.}2^.} has recursion level ${\omega^{\omega^{\varepsilon_02+1}}}$
{1{1,2.2^.}2^.} has recursion level ${\omega^{\omega^{\varepsilon_02+\omega}}}$
{1{1.3^.}2^.} has recursion level ${\omega^{\omega^{\varepsilon_03}}}$
{1{1.1,2^.}2^.} has recursion level ${\omega^{\omega^{\omega^{\varepsilon_0+1}}}}$
{1{1.1.2^.}2^.} has recursion level ${\omega^{\omega^{\omega^{\varepsilon_02}}}}$
{1{1{2^.}2^.}2^.} has recursion level ${\omega^{\omega^{\omega^{\omega^{\varepsilon_0+1}}}}}$
{1{1{1,2^.}2^.}2^.} has recursion level ${\omega^{\omega^{\omega^{\omega^{\varepsilon_0+\omega}}}}}$
{1{1{1.2^.}2^.}2^.} has recursion level ${\omega^{\omega^{\omega^{\omega^{\varepsilon_02}}}}}$
{1{1{1.3^.}2^.}2^.} has recursion level ${\omega^{\omega^{\omega^{\omega^{\varepsilon_03}}}}}$
{1{1{1.1.2^.}2^.}2^.} has recursion level ${\omega^{\omega^{\omega^{\omega^{\omega^{\varepsilon_02}}}}}}$
{1{1{1{2^.}2^.}2^.}2^.} has recursion level ${\omega^{\omega^{\omega^{\omega^{\omega^{\omega^{\varepsilon_0+1}}}}}}}$
{1{1{1{1,2^.}2^.}2^.}2^.} has recursion level ${\omega^{\omega^{\omega^{\omega^{\omega^{\omega^{\varepsilon_0+\omega}}}}}}}$
{1^.2} has recursion level ${\varepsilon_1}$
{2^.2} has recursion level ${\varepsilon_1+1}$
{1,2^.2} has recursion level ${\varepsilon_1+\omega}$
{1.2^.2} has recursion level ${\varepsilon_1+\varepsilon_0}$
{1{1^.2}2^.2} has recursion level ${\varepsilon_12}$
{1{1^.2}3^.2} has recursion level ${\varepsilon_13}$
{1{1^.2}1,2^.2} has recursion level ${\omega^{\varepsilon_1+1}}$
{1{1^.2}1,3^.2} has recursion level ${\omega^{\varepsilon_1+1}2}$
{1{1^.2}1,1,2^.2} has recursion level ${\omega^{\varepsilon_1+2}}$
{1{1^.2}1{2}2^.2} has recursion level ${\omega^{\varepsilon_1+\omega}}$
{1{1^.2}1{1,2}2^.2} has recursion level ${\omega^{\varepsilon_1+\omega^\omega}}$
{1{1^.2}1.2^.2} has recursion level ${\omega^{\varepsilon_1+\varepsilon_0}}$
{1{1^.2}1.1.2^.2} has recursion level ${\omega^{\varepsilon_1+\varepsilon_02}}$
{1{1^.2}1{2^.}2^.2} has recursion level ${\omega^{\varepsilon_1+\omega^{\varepsilon_0+1}}}$
{1{1^.2}1{1.2^.}2^.2} has recursion level ${\omega^{\varepsilon_1+\omega^{\varepsilon_02}}}$
{1{1^.2}1{1^.2}2^.2} has recursion level ${\omega^{\varepsilon_12}}$
{1{2^.2}2^.2} has recursion level ${\omega^{\omega^{\varepsilon_1+1}}}$
{1{2^.2}3^.2} has recursion level ${\omega^{\omega^{\varepsilon_1+1}}2}$
{1{2^.2}1,2^.2} has recursion level ${\omega^{\omega^{\varepsilon_1+1}+1}}$
{1{2^.2}1{2^.2}2^.2} has recursion level ${\omega^{\omega^{\varepsilon_1+1}2}}$
{1{3^.2}2^.2} has recursion level ${\omega^{\omega^{\varepsilon_1+2}}}$
{1{1,2^.2}2^.2} has recursion level ${\omega^{\omega^{\varepsilon_1+\omega}}}$
{1{1{1^.2}2^.2}2^.2} has recursion level ${\omega^{\omega^{\varepsilon_12}}}$
{1{1{1^.2}3^.2}2^.2} has recursion level ${\omega^{\omega^{\varepsilon_13}}}$
{1{1{1^.2}1,2^.2}2^.2} has recursion level ${\omega^{\omega^{\omega^{\varepsilon_1+1}}}}$
{1{1{1^.2}1{1^.2}2^.2}2^.2} has recursion level ${\omega^{\omega^{\omega^{\varepsilon_12}}}}$
{1{1{2^.2}2^.2}2^.2} has recursion level ${\omega^{\omega^{\omega^{\omega^{\varepsilon_1+1}}}}}$
{1{1{1,2^.2}2^.2}2^.2} has recursion level ${\omega^{\omega^{\omega^{\omega^{\varepsilon_1+\omega}}}}}$
{1{1{1{1^.2}2^.2}2^.2}2^.2} has recursion level ${\omega^{\omega^{\omega^{\omega^{\varepsilon_12}}}}}$
{1^.3} has recursion level ${\varepsilon_2}$
{2^.3} has recursion level ${\varepsilon_2+1}$
{1,2^.3} has recursion level ${\varepsilon_2+\omega}$
{1{1^.3}2^.3} has recursion level ${\varepsilon_22}$
{1{1^.3}1,2^.3} has recursion level ${\omega^{\varepsilon_2+1}}$
{1{1^.3}1{1^.3}2^.3} has recursion level ${\omega^{\varepsilon_22}}$
{1{2^.3}2^.3} has recursion level ${\omega^{\omega^{\varepsilon_2+1}}}$
{1{1,2^.3}2^.3} has recursion level ${\omega^{\omega^{\varepsilon_2+\omega}}}$
{1{1{1,2^.3}2^.3}2^.3} has recursion level ${\omega^{\omega^{\omega^{\omega^{\varepsilon_2+\omega}}}}}$
{1^.4} has recursion level ${\varepsilon_3}$
{1^.5} has recursion level ${\varepsilon_4}$

Let f(n) = s(n,n{1^.n}2), then f(n) eventually outgrows all above, and has growth rate ${\varepsilon_\omega}$ .

Expanding arrays

Symbol ` is called grave accent, with ASCII = 96

Now let the grave accent be a separator (add this to the definition of separators), and this will lead to a new kind of arrays, primary expanding arrays. Some simple separators: {1`1} is the comma by rule 2b, {1`2} is the dot, {1`3} is {1^.2}, and {1`k} is {1^.k-1}. The grave accent works as follows: when we meet a non-1 entry immediately after a grave accent, change {1`…1`1`n #} into S_b, where b is the iterator, S₁ is comma and S_i+1 = {1`…1`1 S_i 2`n-1 #}. So for example s(a,b{1`1`3}3) = s(a,b{1`1{1`1…{1`1{1`1,2`2}2`2}…2`2}2`2}2{1`1`3}2) with b-1 2`2}′s.

But the grave accent is just the shorthand for {1^`} where the grave accent is at the left-superscript position of the rbrace. And further, we have {2^`}, {1,2^`}, {1.2^`}, {1{1`1`2}2^`}, {1{1{1,2^`}2}2^`}, {1{1{1{1{1,2^`}2}2^`}2}2^`}, {1`2^`}, {1`1`2^`}, {1{2^`}2^`}, {1{1`2^`}2^`}, {1{1{2^`}2^`}2^`}, etc.

Now the process changes and becomes more complex.

Process

Case B1, B2 and B4 are terminal but case A and B3 are not. And note that red texts imply changes on the array. First start from the 3rd entry.

Case A: If the entry is 1, then you jump to the next entry.
Case B: If the entry n is not 1, look to your left:
- Case B1: If the comma is immediately before you, then
  1. Change the “1,n” into “b,n-1” where n is this non-1 entry and the b is the iterator.
  2. Change all the entries at base layer before them into the base.
  3. The process ends.
- Case B2: If the grave accent (or equivalently, {1^`}) is immediately before you, then
  1. Let t be such that “the grave accent is at layer t”.
  2. Repeat this:
    1. Subtract t by 1.
    2. Let separator B_t be such that it’s at layer t, and the grave accent is inside it.
    3. If t = 1, then break the repeating, or else continue repeating.
  3. Find the maximal t such that A_t hasn’t a grave accent at the left-superscript position of the rbrace.
  4. Let string P and Q be such that B_t = “P ` n Q”
  5. Change B_t into S_b, where b is the iterator, S₁ is comma, and S_i+1 = “P S_i 2 ` n-1 Q”
  6. The process ends.
- Case B3: If a separator K that is neither comma nor grave accent is immediately before you, then
  1. Change the “K n” into “K 2 K n-1”.
  2. Set separator A_t = K, here K is at layer t.
  3. Jump to the first entry of the former K.
- Case B4: If an lbrace is immediately before you, then
  1. Change separator {n #} into string S_b, where b is the iterator, S₁ = “{n-1 #}” and S_i+1 = “S_i 1 {n-1 #}”.
  2. The process ends.

Level comparison

First, all arrays have the lowest and the same level. Then, separators with grave accent at left-superscript always have higher level than those without grave accent. And note that the same separators have the same level. To compare levels of other separators A and B, we follow these steps.

Apply rule 3b to A and B until rule 3b cannot apply any more.
Let A = {a₁A₁a₂A₂…a_k-1A_k-1a_k} and B = {b₁B₁b₂B₂…b_l-1B_l-1b_l}
If k = 1 and l > 1, then lv(A) < lv(B); if k > 1 and l = 1, then lv(A) > lv(B); if k = l = 1, follow step 4; if k > 1 and l > 1, follow step 5 ~ 10
If a₁ < b₁, then lv(A) < lv(B); if a₁ > b₁, then lv(A) > lv(B); if a₁ = b₁, then lv(A) = lv(B)
Let ${M(A)=\{i\in\{1,2,\cdots,k-1\}|\forall j\in\{1,2,\cdots,k-1\}(lv(A_i)\ge lv(A_j))\}}$ , and ${M(B)=\{i\in\{1,2,\cdots,l-1\}|\forall j\in\{1,2,\cdots,l-1\}(lv(B_i)\ge lv(B_j))\}}$ .
If lv(A_maxM(A)) < lv(B_maxM(B)), then lv(A) < lv(B); if lv(A_maxM(A)) > lv(B_maxM(B)), then lv(A) > lv(B); or else –
If |M(A)| < |M(B)|, then lv(A) < lv(B); if |M(A)| > |M(B)|, then lv(A) > lv(B); or else –
Let A = {#₁ A_maxM(A) #₂} and B = {#₃ B_maxM(B) #₄}
If lv({#₂}) < lv({#₄}), then lv(A) < lv(B); if lv({#₂}) > lv({#₄}), then lv(A) > lv(B); or else –
If lv({#₁}) < lv({#₃}), then lv(A) < lv(B); if lv({#₁}) > lv({#₃}), then lv(A) > lv(B); if lv({#₁}) = lv({#₃}), then lv(A) = lv(B)

Explanation

The case B2 is now very complex. And the case B3 also changes. Every time we meet case B3, we need to pull a separator K out, and continue the process inside it. Before the pulling-out, the separators surrounding the grave accent are A_t′s. But when we meet case B2 after all the pulling-out, the separators surrounding the grave accent are B_t′s (and the t is how many layer the A_t′s and B_t′s are at). In step 3 of case B2, the A_t is the innermost separator that hasn’t a grave accent at the left-superscript position of the rbrace. Step 4 and 5 of case B2 mean that the grave accent is pulled out, then the B_t expands into b nests, replacing the grave accent with one less nests of B_t expansion.

For example, s(a,b{1{1`1`3^`}3}2) = s(a,b{1{1`1…{1{1`1{1{1`1,2`2^`}2{1`1`3^`}2}2`2^`}2{1`1`3^`}2}…2`2^`}2{1`1`3^`}2}2) with b-1 {1`1`3^`}′s, and there’re nests of {1{1`1 ____ 2`2^`}2{1`1`3^`}2}. Here A₁ = {1{1`1`3^`}3}, A₂ = {1`1`3^`}, B₁ = {1{1`1`2`2^`}2{1`1`3^`}2}, B₂ = {1`1`2`2^`}, and A₁ is the innermost separator that hasn’t a grave accent at the left-superscript position of the rbrace.

If the grave accent just looks at one layer out, and expands it, things will go wrong. e.g. {1`2^`} will expands into {1{1…{1{1,2^`}2^`}…2^`}2^`}, but if it reduces to {1{1…{1`2^`}…2^`}2^`}, then it expands again, and it can never be solved. So searching for such an A_t is necessary.

Comparison

Here’s the comparison between my array notation and FGH (recursion level). Note again that if separator A has recursion level α > 0, then s(n,n A 2) has growth rate ${\omega^{\omega^\alpha}}$ .

{1`2} has recursion level ${\varepsilon_0}$
{1`3} has recursion level ${\varepsilon_1}$
{1`4} has recursion level ${\varepsilon_2}$
{1`5} has recursion level ${\varepsilon_3}$
{1`1,2} has recursion level ${\varepsilon_\omega}$
{1{1`1,2}2`1,2} has recursion level ${\varepsilon_\omega2}$
{1{1`1,2}1,2`1,2} has recursion level ${\omega^{\varepsilon_\omega+1}}$
{1{1`1,2}1{1`1,2}2`1,2} has recursion level ${\omega^{\varepsilon_\omega2}}$
{1{2`1,2}2`1,2} has recursion level ${\omega^{\omega^{\varepsilon_\omega+1}}}$
{1{1,2`1,2}2`1,2} has recursion level ${\omega^{\omega^{\varepsilon_\omega+\omega}}}$
{1{1{1`1,2}2`1,2}2`1,2} has recursion level ${\omega^{\omega^{\varepsilon_\omega2}}}$
{1{1{1`1,2}1{1`1,2}2`1,2}2`1,2} has recursion level ${\omega^{\omega^{\omega^{\varepsilon_\omega2}}}}$
{1{1{2`1,2}2`1,2}2`1,2} has recursion level ${\omega^{\omega^{\omega^{\omega^{\varepsilon_\omega+1}}}}}$
{1{1{1,2`1,2}2`1,2}2`1,2} has recursion level ${\omega^{\omega^{\omega^{\omega^{\varepsilon_\omega+\omega}}}}}$
{1{1{1{1,2`1,2}2`1,2}2`1,2}2`1,2} has recursion level ${\omega^{\omega^{\omega^{\omega^{\omega^{\omega^{\varepsilon_\omega+\omega}}}}}}}$
{1`2,2} has recursion level ${\varepsilon_{\omega+1}}$
{1`3,2} has recursion level ${\varepsilon_{\omega+2}}$
{1`1,3} has recursion level ${\varepsilon_{\omega2}}$
{1`1,1,2} has recursion level ${\varepsilon_{\omega^2}}$
{1`1{2}2} has recursion level ${\varepsilon_{\omega^\omega}}$
{1`1{1,2}2} has recursion level ${\varepsilon_{\omega^{\omega^\omega}}}$
{1`1{1{1,2}2}2} has recursion level ${\varepsilon_{\omega^{\omega^{\omega^{\omega^\omega}}}}}$
{1`1.2} = {1`1{1`2}2} has recursion level ${\varepsilon_{\varepsilon_0}}$
{1`2.2} has recursion level ${\varepsilon_{\varepsilon_0+1}}$
{1`1,2.2} has recursion level ${\varepsilon_{\varepsilon_0+\omega}}$
{1`1{1,2}2.2} has recursion level ${\varepsilon_{\varepsilon_0+\omega^{\omega^\omega}}}$
{1`1.3} = {1`1{1`2}3} has recursion level ${\varepsilon_{\varepsilon_02}}$
{1`1.4} has recursion level ${\varepsilon_{\varepsilon_03}}$
{1`1.1,2} has recursion level ${\varepsilon_{\omega^{\varepsilon_0+1}}}$
{1`1.1,3} has recursion level ${\varepsilon_{\omega^{\varepsilon_0+1}2}}$
{1`1.1,1,2} has recursion level ${\varepsilon_{\omega^{\varepsilon_0+2}}}$
{1`1.1{2}2} has recursion level ${\varepsilon_{\omega^{\varepsilon_0+\omega}}}$
{1`1.1.2} has recursion level ${\varepsilon_{\omega^{\varepsilon_02}}}$
{1`1{2`2}2} has recursion level ${\varepsilon_{\omega^{\omega^{\varepsilon_0+1}}}}$
{1`1{3`2}2} has recursion level ${\varepsilon_{\omega^{\omega^{\varepsilon_0+2}}}}$
{1`1{1,2`2}2} has recursion level ${\varepsilon_{\omega^{\omega^{\varepsilon_0+\omega}}}}$
{1`1{1.2`2}2} has recursion level ${\varepsilon_{\omega^{\omega^{\varepsilon_02}}}}$
{1`1{1.1.2`2}2} has recursion level ${\varepsilon_{\omega^{\omega^{\omega^{\varepsilon_02}}}}}$
{1`1{1{2`2}2`2}2} has recursion level ${\varepsilon_{\omega^{\omega^{\omega^{\omega^{\varepsilon_0+1}}}}}}$
{1`1{1{1,2`2}2`2}2} has recursion level ${\varepsilon_{\omega^{\omega^{\omega^{\omega^{\varepsilon_0+\omega}}}}}}$
{1`1{1`3}2} has recursion level ${\varepsilon_{\varepsilon_1}}$
{1`1{1`3}3} has recursion level ${\varepsilon_{\varepsilon_12}}$
{1`1{1`3}1{1`3}2} has recursion level ${\varepsilon_{\omega^{\varepsilon_12}}}$
{1`1{2`3}2} has recursion level ${\varepsilon_{\omega^{\omega^{\varepsilon_1+1}}}}$
{1`1{1,2`3}2} has recursion level ${\varepsilon_{\omega^{\omega^{\varepsilon_1+\omega}}}}$
{1`1{1{1`3}2`3}2} has recursion level ${\varepsilon_{\omega^{\omega^{\varepsilon_12}}}}$
{1`1{1`4}2} has recursion level ${\varepsilon_{\varepsilon_2}}$
{1`1{1`5}2} has recursion level ${\varepsilon_{\varepsilon_3}}$
{1`1{1`1,2}2} has recursion level ${\varepsilon_{\varepsilon_\omega}}$
{1`2{1`1,2}2} has recursion level ${\varepsilon_{\varepsilon_\omega+1}}$
{1`1{1`1,2}3} has recursion level ${\varepsilon_{\varepsilon_\omega2}}$
{1`1{1`2,2}2} has recursion level ${\varepsilon_{\varepsilon_{\omega+1}}}$
{1`1{1`1,3}2} has recursion level ${\varepsilon_{\varepsilon_{\omega2}}}$
{1`1{1`1,1,2}2} has recursion level ${\varepsilon_{\varepsilon_{\omega^2}}}$
{1`1{1`1{2}2}2} has recursion level ${\varepsilon_{\varepsilon_{\omega^\omega}}}$
{1`1{1`1.2}2} = {1`1{1`1{1`2}2}2} has recursion level ${\varepsilon_{\varepsilon_{\varepsilon_0}}}$
{1`1{1`1.2}3} has recursion level ${\varepsilon_{\varepsilon_{\varepsilon_0}2}}$
{1`1{1`2.2}2} has recursion level ${\varepsilon_{\varepsilon_{\varepsilon_0+1}}}$
{1`1{1`1.3}2} has recursion level ${\varepsilon_{\varepsilon_{\varepsilon_02}}}$
{1`1{1`1{1`3}2}2} has recursion level ${\varepsilon_{\varepsilon_{\varepsilon_1}}}$
{1`1{1`1{1`1,2}2}2} has recursion level ${\varepsilon_{\varepsilon_{\varepsilon_\omega}}}$
{1`1{1`1{1`1.2}2}2} has recursion level ${\varepsilon_{\varepsilon_{\varepsilon_{\varepsilon_0}}}}$
{1`1{1`1{1`1{1`1,2}2}2}2} has recursion level ${\varepsilon_{\varepsilon_{\varepsilon_{\varepsilon_\omega}}}}$
{1`1`2} has recursion level ${\varphi(2,0)}$
{2`1`2} has recursion level ${\varphi(2,0)+1}$
{1{1`1`2}2`1`2} has recursion level ${\varphi(2,0)2}$
{1{1`1`2}1{1`1`2}2`1`2} has recursion level ${\omega^{\varphi(2,0)2}}$
{1{2`1`2}2`1`2} has recursion level ${\omega^{\omega^{\varphi(2,0)+1}}}$
{1{1{1`1`2}2`1`2}2`1`2} has recursion level ${\omega^{\omega^{\varphi(2,0)2}}}$
{1`2`2} has recursion level ${\varepsilon_{\varphi(2,0)+1}}$
{1`3`2} has recursion level ${\varepsilon_{\varphi(2,0)+2}}$
{1`1.2`2} has recursion level ${\varepsilon_{\varphi(2,0)+\varepsilon_0}}$
{1`1{1`1`2}2`2} has recursion level ${\varepsilon_{\varphi(2,0)2}}$
{1`2{1`1`2}2`2} has recursion level ${\varepsilon_{\varphi(2,0)2+1}}$
{1`1{1`1`2}3`2} has recursion level ${\varepsilon_{\varphi(2,0)3}}$
{1`1{1`1`2}1,2`2} has recursion level ${\varepsilon_{\omega^{\varphi(2,0)+1}}}$
{1`1{1`1`2}1{1`1`2}2`2} has recursion level ${\varepsilon_{\omega^{\varphi(2,0)2}}}$
{1`1{2`1`2}2`2} has recursion level ${\varepsilon_{\omega^{\omega^{\varphi(2,0)+1}}}}$
{1`1{1{1`1`2}2`1`2}2`2} has recursion level ${\varepsilon_{\omega^{\omega^{\varphi(2,0)2}}}}$
{1`1{1{1`1`2}3`1`2}2`2} has recursion level ${\varepsilon_{\omega^{\omega^{\varphi(2,0)3}}}}$
{1`1{1{1`1`2}1{1`1`2}2`1`2}2`2} has recursion level ${\varepsilon_{\omega^{\omega^{\omega^{\varphi(2,0)2}}}}}$
{1`1{1{2`1`2}2`1`2}2`2} has recursion level ${\varepsilon_{\omega^{\omega^{\omega^{\omega^{\varphi(2,0)+1}}}}}}$
{1`1{1{1{1`1`2}2`1`2}2`1`2}2`2} has recursion level ${\varepsilon_{\omega^{\omega^{\omega^{\omega^{\varphi(2,0)2}}}}}}$
{1`1{1`2`2}2`2} has recursion level ${\varepsilon_{\varepsilon_{\varphi(2,0)+1}}}$
{1`2{1`2`2}2`2} has recursion level ${\varepsilon_{\varepsilon_{\varphi(2,0)+1}+1}}$
{1`1{1`2`2}3`2} has recursion level ${\varepsilon_{\varepsilon_{\varphi(2,0)+1}2}}$
{1`1{1`2`2}1{1`2`2}2`2} has recursion level ${\varepsilon_{\omega^{\varepsilon_{\varphi(2,0)+1}2}}}$
{1`1{2`2`2}2`2} has recursion level ${\varepsilon_{\omega^{\omega^{\varepsilon_{\varphi(2,0)+1}+1}}}}$
{1`1{1`3`2}2`2} has recursion level ${\varepsilon_{\varepsilon_{\varphi(2,0)+2}}}$
{1`1{1`1,2`2}2`2} has recursion level ${\varepsilon_{\varepsilon_{\varphi(2,0)+\omega}}}$
{1`1{1`1{1`1`2}2`2}2`2} has recursion level ${\varepsilon_{\varepsilon_{\varphi(2,0)2}}}$
{1`1{1`1{1`1`2}1{1`1`2}2`2}2`2} has recursion level ${\varepsilon_{\varepsilon_{\omega^{\varphi(2,0)2}}}}$
{1`1{1`1{1{1`1`2}2`1`2}2`2}2`2} has recursion level ${\varepsilon_{\varepsilon_{\omega^{\omega^{\varphi(2,0)2}}}}}$
{1`1{1`1{1`2`2}2`2}2`2} has recursion level ${\varepsilon_{\varepsilon_{\varepsilon_{\varphi(2,0)+1}}}}$
{1`1{1`1{1`3`2}2`2}2`2} has recursion level ${\varepsilon_{\varepsilon_{\varepsilon_{\varphi(2,0)+2}}}}$
{1`1{1`1{1`1,2`2}2`2}2`2} has recursion level ${\varepsilon_{\varepsilon_{\varepsilon_{\varphi(2,0)+\omega}}}}$
{1`1{1`1{1`1{1`1`2}2`2}2`2}2`2} has recursion level ${\varepsilon_{\varepsilon_{\varepsilon_{\varphi(2,0)2}}}}$
{1`1{1`1{1`1{1`1,2`2}2`2}2`2}2`2} has recursion level ${\varepsilon_{\varepsilon_{\varepsilon_{\varepsilon_{\varphi(2,0)+\omega}}}}}$
{1`1`3} has recursion level ${\varphi(2,1)}$
{1`2`3} has recursion level ${\varepsilon_{\varphi(2,1)+1}}$
{1`1{1`1`3}2`3} has recursion level ${\varepsilon_{\varphi(2,1)2}}$
{1`1{1`1`3}1{1`1`3}2`3} has recursion level ${\varepsilon_{\omega^{\varphi(2,1)2}}}$
{1`1{2`1`3}2`3} has recursion level ${\varepsilon_{\omega^{\omega^{\varphi(2,1)+1}}}}$
{1`1{1{1`1`3}2`1`3}2`3} has recursion level ${\varepsilon_{\omega^{\omega^{\varphi(2,1)2}}}}$
{1`1{1`2`3}2`3} has recursion level ${\varepsilon_{\varepsilon_{\varphi(2,1)+1}}}$
{1`1{1`1{1`1`3}2`3}2`3} has recursion level ${\varepsilon_{\varepsilon_{\varphi(2,1)2}}}$
{1`1{1`1{1`2`3}2`3}2`3} has recursion level ${\varepsilon_{\varepsilon_{\varepsilon_{\varphi(2,1)+1}}}}$
{1`1{1`1{1`1{1`1`3}2`3}2`3}2`3} has recursion level ${\varepsilon_{\varepsilon_{\varepsilon_{\varphi(2,1)2}}}}$
{1`1`4} has recursion level ${\varphi(2,2)}$
{1`1`5} has recursion level ${\varphi(2,3)}$
{1`1`1,2} has recursion level ${\varphi(2,\omega)}$
{1`1`1.2} has recursion level ${\varphi(2,\varepsilon_0)}$
{1`1`1{1`1`2}2} has recursion level ${\varphi(2,\varphi(2,0))}$
{1`1`2{1`1`2}2} has recursion level ${\varphi(2,\varphi(2,0)+1)}$
{1`1`1{1`1`2}3} has recursion level ${\varphi(2,\varphi(2,0)2)}$
{1`1`1{1`1`2}1{1`1`2}2} has recursion level ${\varphi(2,\omega^{\varphi(2,0)2})}$
{1`1`1{2`1`2}2} has recursion level ${\varphi(2,\omega^{\omega^{\varphi(2,0)+1}})}$
{1`1`1{1{1`1`2}2`1`2}2} has recursion level ${\varphi(2,\omega^{\omega^{\varphi(2,0)2}})}$
{1`1`1{1`2`2}2} has recursion level ${\varphi(2,\varepsilon_{\varphi(2,0)+1})}$
{1`1`1{1`1{1`1`2}2`2}2} has recursion level ${\varphi(2,\varepsilon_{\varphi(2,0)2})}$
{1`1`1{1`1{1`2`2}2`2}2} has recursion level ${\varphi(2,\varepsilon_{\varepsilon_{\varphi(2,0)+1}})}$
{1`1`1{1`1{1`1{1`1`2}2`2}2`2}2} has recursion level ${\varphi(2,\varepsilon_{\varepsilon_{\varphi(2,0)2}})}$
{1`1`1{1`1`3}2} has recursion level ${\varphi(2,\varphi(2,1))}$
{1`1`1{1`1`4}2} has recursion level ${\varphi(2,\varphi(2,2))}$
{1`1`1{1`1`1,2}2} has recursion level ${\varphi(2,\varphi(2,\omega))}$
{1`1`1{1`1`1{1`1`2}2}2} has recursion level ${\varphi(2,\varphi(2,\varphi(2,0)))}$
{1`1`2{1`1`1{1`1`2}2}2} has recursion level ${\varphi(2,\varphi(2,\varphi(2,0))+1)}$
{1`1`1{1`1`1{1`1`2}2}3} has recursion level ${\varphi(2,\varphi(2,\varphi(2,0))2)}$
{1`1`1{1`1`2{1`1`2}2}2} has recursion level ${\varphi(2,\varphi(2,\varphi(2,0)+1))}$
{1`1`1{1`1`1{1`1`2}3}2} has recursion level ${\varphi(2,\varphi(2,\varphi(2,0)2))}$
{1`1`1{1`1`1{1`1`3}2}2} has recursion level ${\varphi(2,\varphi(2,\varphi(2,1)))}$
{1`1`1{1`1`1{1`1`1{1`1`2}2}2}2} has recursion level ${\varphi(2,\varphi(2,\varphi(2,\varphi(2,0))))}$
{1`1`1{1`1`1{1`1`1{1`1`1{1`1`2}2}2}2}2} has recursion level ${\varphi(2,\varphi(2,\varphi(2,\varphi(2,\varphi(2,0)))))}$
{1`1`1`2} has recursion level ${\varphi(3,0)}$
{1`2`1`2} has recursion level ${\varepsilon_{\varphi(3,0)+1}}$
{1`1{1`1`1`2}2`1`2} has recursion level ${\varepsilon_{\varphi(3,0)2}}$
{1`1{1`2`1`2}2`1`2} has recursion level ${\varepsilon_{\varepsilon_{\varphi(3,0)+1}}}$
{1`1{1`1{1`1`1`2}2`1`2}2`1`2} has recursion level ${\varepsilon_{\varepsilon_{\varphi(3,0)2}}}$
{1`1`2`2} has recursion level ${\varphi(2,\varphi(3,0)+1)}$
{1`1`1{1`1`1`2}2`2} has recursion level ${\varphi(2,\varphi(3,0)2)}$
{1`1`1{1`1`2`2}2`2} has recursion level ${\varphi(2,\varphi(2,\varphi(3,0)+1))}$
{1`1`1{1`1`1{1`1`1`2}2`2}2`2} has recursion level ${\varphi(2,\varphi(2,\varphi(3,0)2))}$
{1`1`1`3} has recursion level ${\varphi(3,1)}$
{1`1`1`4} has recursion level ${\varphi(3,2)}$
{1`1`1`1,2} has recursion level ${\varphi(3,\omega)}$
{1`1`1`1{1`1`1`2}2} has recursion level ${\varphi(3,\varphi(3,0))}$
{1`1`1`1{1`1`1`3}2} has recursion level ${\varphi(3,\varphi(3,1))}$
{1`1`1`1{1`1`1`1{1`1`1`2}2}2} has recursion level ${\varphi(3,\varphi(3,\varphi(3,0)))}$
{1`1`1`1`2} has recursion level ${\varphi(4,0)}$
{1`1`1`1`1`2} has recursion level ${\varphi(5,0)}$
{1{2^`}2} has recursion level ${\varphi(\omega,0)}$ , so s(n,n{1{2^`}2}2) reaches the limit of linear arrays in hyperfactorial array notation (i.e. n![1,1,…1,1,2] with n 1’s)
{1`2{2^`}2} has recursion level ${\varepsilon_{\varphi(\omega,0)+1}}$
{1`1`2{2^`}2} has recursion level ${\varphi(2,\varphi(\omega,0)+1)}$
{1`1`1`2{2^`}2} has recursion level ${\varphi(3,\varphi(\omega,0)+1)}$
{1{2^`}3} has recursion level ${\varphi(\omega,1)}$
{1{2^`}4} has recursion level ${\varphi(\omega,2)}$
{1{2^`}1,2} has recursion level ${\varphi(\omega,\omega)}$
{1{2^`}1.2} has recursion level ${\varphi(\omega,\varepsilon_0)}$
{1{2^`}1{1`3}2} has recursion level ${\varphi(\omega,\varepsilon_1)}$
{1{2^`}1{1`1`2}2} has recursion level ${\varphi(\omega,\varphi(2,0))}$
{1{2^`}1{1`1`1`2}2} has recursion level ${\varphi(\omega,\varphi(3,0))}$
{1{2^`}1{1{2^`}2}2} has recursion level ${\varphi(\omega,\varphi(\omega,0))}$
{1{2^`}2{1{2^`}2}2} has recursion level ${\varphi(\omega,\varphi(\omega,0)+1)}$
{1{2^`}1{1{2^`}2}3} has recursion level ${\varphi(\omega,\varphi(\omega,0)2)}$
{1{2^`}1{1`2{2^`}2}2} has recursion level ${\varphi(\omega,\varepsilon_{\varphi(\omega,0)+1})}$
{1{2^`}1{1`1`2{2^`}2}2} has recursion level ${\varphi(\omega,\varphi(2,\varphi(\omega,0)+1))}$
{1{2^`}1{1{2^`}3}2} has recursion level ${\varphi(\omega,\varphi(\omega,1))}$
{1{2^`}1{1{2^`}1,2}2} has recursion level ${\varphi(\omega,\varphi(\omega,\omega))}$
{1{2^`}1{1{2^`}1{1{2^`}2}2}2} has recursion level ${\varphi(\omega,\varphi(\omega,\varphi(\omega,0)))}$
{1{2^`}1{1{2^`}1{1{2^`}3}2}2} has recursion level ${\varphi(\omega,\varphi(\omega,\varphi(\omega,1)))}$
{1{2^`}1{1{2^`}1{1{2^`}1{1{2^`}2}2}2}2} has recursion level ${\varphi(\omega,\varphi(\omega,\varphi(\omega,\varphi(\omega,0))))}$
{1{2^`}1`2} has recursion level ${\varphi(\omega+1,0)}$
{1{2^`}2`2} has recursion level ${\varphi(\omega,\varphi(\omega+1,0)+1)}$
{1{2^`}1`3} has recursion level ${\varphi(\omega+1,1)}$
{1{2^`}1`1,2} has recursion level ${\varphi(\omega+1,\omega)}$
{1{2^`}1`1{1{2^`}1`2}2} has recursion level ${\varphi(\omega+1,\varphi(\omega+1,0))}$
{1{2^`}1`1{1{2^`}1`3}2} has recursion level ${\varphi(\omega+1,\varphi(\omega+1,1))}$
{1{2^`}1`1{1{2^`}1`1{1{2^`}1`2}2}2} has recursion level ${\varphi(\omega+1,\varphi(\omega+1,\varphi(\omega+1,0)))}$
{1{2^`}1`1`2} has recursion level ${\varphi(\omega+2,0)}$
{1{2^`}1`1`1`2} has recursion level ${\varphi(\omega+3,0)}$
{1{2^`}1{2^`}2} has recursion level ${\varphi(\omega2,0)}$
{1{2^`}1{2^`}3} has recursion level ${\varphi(\omega2,1)}$
{1{2^`}1{2^`}1`2} has recursion level ${\varphi(\omega2+1,0)}$
{1{2^`}1{2^`}1{2^`}2} has recursion level ${\varphi(\omega3,0)}$
{1{2^`}1{2^`}1{2^`}1{2^`}2} has recursion level ${\varphi(\omega4,0)}$
{1{3^`}2} has recursion level ${\varphi(\omega^2,0)}$
{1{4^`}2} has recursion level ${\varphi(\omega^3,0)}$
{1{1,2^`}2} has recursion level ${\varphi(\omega^\omega,0)}$
{1{1{2}2^`}2} has recursion level ${\varphi(\omega^{\omega^\omega},0)}$
{1{1{1,2}2^`}2} has recursion level ${\varphi(\omega^{\omega^{\omega^\omega}},0)}$
{1{1.2^`}2} has recursion level ${\varphi(\varepsilon_0,0)}$
{1{1{1`3}2^`}2} has recursion level ${\varphi(\varepsilon_1,0)}$
{1{1{1`1`2}2^`}2} has recursion level ${\varphi(\varphi(2,0),0)}$
{1{1{1`1`1`2}2^`}2} has recursion level ${\varphi(\varphi(3,0),0)}$
{1{1{1{2^`}2}2^`}2} has recursion level ${\varphi(\varphi(\omega,0),0)}$
{1{1{1{2^`}1`2}2^`}2} has recursion level ${\varphi(\varphi(\omega+1,0),0)}$
{1{1{1{2^`}1{2^`}2}2^`}2} has recursion level ${\varphi(\varphi(\omega2,0),0)}$
{1{1{1{3^`}2}2^`}2} has recursion level ${\varphi(\varphi(\omega^2,0),0)}$
{1{1{1{1,2^`}2}2^`}2} has recursion level ${\varphi(\varphi(\omega^\omega,0),0)}$
{1{1{1{1.2^`}2}2^`}2} has recursion level ${\varphi(\varphi(\varepsilon_0,0),0)}$
{1{1{1{1{1`1`2}2^`}2}2^`}2} has recursion level ${\varphi(\varphi(\varphi(2,0),0),0)}$
{1{1{1{1{1{2^`}2}2^`}2}2^`}2} has recursion level ${\varphi(\varphi(\varphi(\omega,0),0),0)}$
{1{1{1{1{1{1,2^`}2}2^`}2}2^`}2} has recursion level ${\varphi(\varphi(\varphi(\omega^\omega,0),0),0)}$
{1{1{1{1{1{1{1{1,2^`}2}2^`}2}2^`}2}2^`}2} has recursion level ${\varphi(\varphi(\varphi(\varphi(\omega^\omega,0),0),0),0)}$
{1{1`2^`}2} has recursion level ${\Gamma_0}$ ( ${\Gamma_0[0]=0}$ and ${\Gamma_0[n+1]=\varphi(\Gamma_0[n],0)}$ ), so s(n,n{1{1`2^`}2}2) eventually outgrows all functions provably recursive in arithmetical transfinite recursion (ATR₀)
{1`2{1`2^`}2} has recursion level ${\varepsilon_{\Gamma_0+1}}$
{1`1`2{1`2^`}2} has recursion level ${\varphi(2,\Gamma_0+1)}$
{1{2^`}2{1`2^`}2} has recursion level ${\varphi(\omega,\Gamma_0+1)}$
{1{1,2^`}2{1`2^`}2} has recursion level ${\varphi(\omega^\omega,\Gamma_0+1)}$
{1{1.2^`}2{1`2^`}2} has recursion level ${\varphi(\varepsilon_0,\Gamma_0+1)}$
{1{1{1`1`2}2^`}2{1`2^`}2} has recursion level ${\varphi(\varphi(2,0),\Gamma_0+1)}$
{1{1{1{2^`}2}2^`}2{1`2^`}2} has recursion level ${\varphi(\varphi(\omega,0),\Gamma_0+1)}$
{1{1{1{1,2^`}2}2^`}2{1`2^`}2} has recursion level ${\varphi(\varphi(\omega^\omega,0),\Gamma_0+1)}$
{1{1{1{1.2^`}2}2^`}2{1`2^`}2} has recursion level ${\varphi(\varphi(\varepsilon_0,0),\Gamma_0+1)}$
{1{1{1{1{1`1`2}2^`}2}2^`}2{1`2^`}2} has recursion level ${\varphi(\varphi(\varphi(2,0),0),\Gamma_0+1)}$
{1{1{1{1{1{2^`}2}2^`}2}2^`}2{1`2^`}2} has recursion level ${\varphi(\varphi(\varphi(\omega,0),0),\Gamma_0+1)}$
{1{1{1{1{1{1,2^`}2}2^`}2}2^`}2{1`2^`}2} has recursion level ${\varphi(\varphi(\varphi(\omega^\omega,0),0),\Gamma_0+1)}$
{1{1{1{1`2^`}2}2^`}2{1`2^`}2} has recursion level ${\varphi(\Gamma_0,1)}$
{1`2{1{1{1`2^`}2}2^`}2{1`2^`}2} has recursion level ${\varepsilon_{\varphi(\Gamma_0,1)+1}}$
{1{2^`}2{1{1{1`2^`}2}2^`}2{1`2^`}2} has recursion level ${\varphi(\omega,\varphi(\Gamma_0,1)+1)}$
{1{1,2^`}2{1{1{1`2^`}2}2^`}2{1`2^`}2} has recursion level ${\varphi(\omega^\omega,\varphi(\Gamma_0,1)+1)}$
{1{1{1`2}2^`}2{1{1{1`2^`}2}2^`}2{1`2^`}2} has recursion level ${\varphi(\varepsilon_0,\varphi(\Gamma_0,1)+1)}$
{1{1{1{2^`}2}2^`}2{1{1{1`2^`}2}2^`}2{1`2^`}2} has recursion level ${\varphi(\varphi(\omega,0),\varphi(\Gamma_0,1)+1)}$
{1{1{1{1,2^`}2}2^`}2{1{1{1`2^`}2}2^`}2{1`2^`}2} has recursion level ${\varphi(\varphi(\omega^\omega,0),\varphi(\Gamma_0,1)+1)}$
{1{1{1{1{1{1,2^`}2}2^`}2}2^`}2{1{1{1`2^`}2}2^`}2{1`2^`}2} has recursion level ${\varphi(\varphi(\varphi(\omega^\omega,0),0),\varphi(\Gamma_0,1)+1)}$
{1{1{1{1`2^`}2}2^`}3{1`2^`}2} has recursion level ${\varphi(\Gamma_0,2)}$
{1{1{1{1`2^`}2}2^`}1,2{1`2^`}2} has recursion level ${\varphi(\Gamma_0,\omega)}$
{1{1{1{1`2^`}2}2^`}1{1{1`2^`}2}2{1`2^`}2} has recursion level ${\varphi(\Gamma_0,\Gamma_0)}$
{1{1{1{1`2^`}2}2^`}1{1{1{1{1`2^`}2}2^`}2{1`2^`}2}2{1`2^`}2} has recursion level ${\varphi(\Gamma_0,\varphi(\Gamma_0,1))}$
{1{1{1{1`2^`}2}2^`}1{1{1{1{1`2^`}2}2^`}3{1`2^`}2}2{1`2^`}2} has recursion level ${\varphi(\Gamma_0,\varphi(\Gamma_0,2))}$
{1{1{1{1`2^`}2}2^`}1{1{1{1{1`2^`}2}2^`}1{1{1`2^`}2}2{1`2^`}2}2{1`2^`}2} has recursion level ${\varphi(\Gamma_0,\varphi(\Gamma_0,\Gamma_0))}$
{1{1{1{1`2^`}2}2^`}1`2{1`2^`}2} has recursion level ${\varphi(\Gamma_0+1,0)}$
{1{1{1{1`2^`}2}2^`}1`3{1`2^`}2} has recursion level ${\varphi(\Gamma_0+1,1)}$
{1{1{1{1`2^`}2}2^`}1`1`2{1`2^`}2} has recursion level ${\varphi(\Gamma_0+2,0)}$
{1{1{1{1`2^`}2}2^`}1{2^`}2{1`2^`}2} has recursion level ${\varphi(\Gamma_0+\omega,0)}$
{1{1{1{1`2^`}2}2^`}1{1,2^`}2{1`2^`}2} has recursion level ${\varphi(\Gamma_0+\omega^\omega,0)}$
{1{1{1{1`2^`}2}2^`}1{1.2^`}2{1`2^`}2} has recursion level ${\varphi(\Gamma_0+\varepsilon_0,0)}$
{1{1{1{1`2^`}2}2^`}1{1{1{2^`}2}2^`}2{1`2^`}2} has recursion level ${\varphi(\Gamma_0+\varphi(\omega,0),0)}$
{1{1{1{1`2^`}2}2^`}1{1{1{1,2^`}2}2^`}2{1`2^`}2} has recursion level ${\varphi(\Gamma_0+\varphi(\omega^\omega,0),0)}$
{1{1{1{1`2^`}2}2^`}1{1{1{1{1{1,2^`}2}2^`}2}2^`}2{1`2^`}2} has recursion level ${\varphi(\Gamma_0+\varphi(\varphi(\omega^\omega,0),0),0)}$
{1{1{1{1`2^`}2}2^`}1{1{1{1`2^`}2}2^`}2{1`2^`}2} has recursion level ${\varphi(\Gamma_02,0)}$
{1{1{1{1`2^`}2}2^`}1{1{1{1`2^`}2}2^`}1`2{1`2^`}2} has recursion level ${\varphi(\Gamma_02+1,0)}$
{1{1{1{1`2^`}2}2^`}1{1{1{1`2^`}2}2^`}1{1{1{1`2^`}2}2^`}2{1`2^`}2} has recursion level ${\varphi(\Gamma_03,0)}$
{1{2{1{1`2^`}2}2^`}2{1`2^`}2} has recursion level ${\varphi(\omega^{\Gamma_0+1},0)}$
{1{1,2{1{1`2^`}2}2^`}2{1`2^`}2} has recursion level ${\varphi(\omega^{\Gamma_0+\omega},0)}$
{1{1.2{1{1`2^`}2}2^`}2{1`2^`}2} has recursion level ${\varphi(\omega^{\Gamma_0+\varepsilon_0},0)}$
{1{1{1{2^`}2}2{1{1`2^`}2}2^`}2{1`2^`}2} has recursion level ${\varphi(\omega^{\Gamma_0+\varphi(\omega,0)},0)}$
{1{1{1{1,2^`}2}2{1{1`2^`}2}2^`}2{1`2^`}2} has recursion level ${\varphi(\omega^{\Gamma_0+\varphi(\omega^\omega,0)},0)}$
{1{1{1{1{1{1,2^`}2}2^`}2}2{1{1`2^`}2}2^`}2{1`2^`}2} has recursion level ${\varphi(\omega^{\Gamma_0+\varphi(\varphi(\omega^\omega,0),0)},0)}$
{1{1{1{1`2^`}2}3^`}2{1`2^`}2} has recursion level ${\varphi(\omega^{\Gamma_02},0)}$
{1{1{1{1`2^`}2}4^`}2{1`2^`}2} has recursion level ${\varphi(\omega^{\Gamma_03},0)}$
{1{1{1{1`2^`}2}1{1{1`2^`}2}2^`}2{1`2^`}2} has recursion level ${\varphi(\omega^{\omega^{\Gamma_02}},0)}$
{1{1{2{1`2^`}2}2^`}2{1`2^`}2} has recursion level ${\varphi(\omega^{\omega^{\omega^{\Gamma_0+1}}},0)}$
{1{1{3{1`2^`}2}2^`}2{1`2^`}2} has recursion level ${\varphi(\omega^{\omega^{\omega^{\Gamma_0+2}}},0)}$
{1{1{1{1{1`2^`}2}2{1`2^`}2}2^`}2{1`2^`}2} has recursion level ${\varphi(\omega^{\omega^{\omega^{\Gamma_02}}},0)}$
{1{1{1{1{1`2^`}2}1{1{1`2^`}2}2{1`2^`}2}2^`}2{1`2^`}2} has recursion level ${\varphi(\omega^{\omega^{\omega^{\omega^{\Gamma_02}}}},0)}$
{1{1{1{2{1`2^`}2}2{1`2^`}2}2^`}2{1`2^`}2} has recursion level ${\varphi(\omega^{\omega^{\omega^{\omega^{\omega^{\Gamma_0+1}}}}},0)}$
{1{1{1{1{1{1`2^`}2}2{1`2^`}2}2{1`2^`}2}2^`}2{1`2^`}2} has recursion level ${\varphi(\omega^{\omega^{\omega^{\omega^{\omega^{\Gamma_02}}}}},0)}$
{1{1{1`2{1`2^`}2}2^`}2{1`2^`}2} has recursion level ${\varphi(\varepsilon_{\Gamma_0+1},0)}$
{1{1{1{2^`}2{1`2^`}2}2^`}2{1`2^`}2} has recursion level ${\varphi(\varphi(\omega,\Gamma_0+1),0)}$
{1{1{1{1,2^`}2{1`2^`}2}2^`}2{1`2^`}2} has recursion level ${\varphi(\varphi(\omega^\omega,\Gamma_0+1),0)}$
{1{1{1{1{1{1,2^`}2}2^`}2{1`2^`}2}2^`}2{1`2^`}2} has recursion level ${\varphi(\varphi(\varphi(\omega^\omega,0),\Gamma_0+1),0)}$
{1{1{1{1{1{1`2^`}2}2^`}2{1`2^`}2}2^`}2{1`2^`}2} has recursion level ${\varphi(\varphi(\Gamma_0,1),0)}$
{1{1{1{1{1{1`2^`}2}2^`}2{1`2^`}2}2^`}3{1`2^`}2} has recursion level ${\varphi(\varphi(\Gamma_0,1),1)}$
{1{2{1{1{1{1`2^`}2}2^`}2{1`2^`}2}2^`}2{1`2^`}2} has recursion level ${\varphi(\omega^{\varphi(\Gamma_0,1)+1},0)}$
{1{1{1{1{1{1`2^`}2}2^`}2{1`2^`}2}3^`}2{1`2^`}2} has recursion level ${\varphi(\omega^{\varphi(\Gamma_0,1)2},0)}$
{1{1{2{1{1{1`2^`}2}2^`}2{1`2^`}2}2^`}2{1`2^`}2} has recursion level ${\varphi(\omega^{\omega^{\varphi(\Gamma_0,1)+1}},0)}$
{1{1{1{1{1{1`2^`}2}2^`}3{1`2^`}2}2^`}2{1`2^`}2} has recursion level ${\varphi(\varphi(\Gamma_0,2),0)}$
{1{1{1{2{1{1`2^`}2}2^`}2{1`2^`}2}2^`}2{1`2^`}2} has recursion level ${\varphi(\varphi(\omega^{\Gamma_0+1},0),0)}$
{1{1{1{1{1{1`2^`}2}3^`}2{1`2^`}2}2^`}2{1`2^`}2} has recursion level ${\varphi(\varphi(\omega^{\Gamma_02},0),0)}$
{1{1{1{1{2{1`2^`}2}2^`}2{1`2^`}2}2^`}2{1`2^`}2} has recursion level ${\varphi(\varphi(\omega^{\omega^{\Gamma_0+1}},0),0)}$
{1{1{1{1{1`2{1`2^`}2}2^`}2{1`2^`}2}2^`}2{1`2^`}2} has recursion level ${\varphi(\varphi(\varepsilon_{\Gamma_0+1},0),0)}$
{1{1{1{1{1{1,2^`}2{1`2^`}2}2^`}2{1`2^`}2}2^`}2{1`2^`}2} has recursion level ${\varphi(\varphi(\varphi(\omega^\omega,\Gamma_0+1),0),0)}$
{1{1{1{1{1{1{1{1`2^`}2}2^`}2{1`2^`}2}2^`}2{1`2^`}2}2^`}2{1`2^`}2} has recursion level ${\varphi(\varphi(\varphi(\Gamma_0,1),0),0)}$
{1{1{1{1{1{1{1{1{1{1`2^`}2}2^`}2{1`2^`}2}2^`}2{1`2^`}2}2^`}2{1`2^`}2}2^`}2{1`2^`}2} has recursion level ${\varphi(\varphi(\varphi(\varphi(\Gamma_0,1),0),0),0)}$
{1{1`2^`}3} has recursion level ${\Gamma_1}$ ( ${\Gamma_{\alpha+1}[0]=\Gamma_\alpha+1}$ and ${\Gamma_{\alpha+1}[n+1]=\varphi(\Gamma_{\alpha+1}[n],0)}$ ; and for limit α the fundamental sequences are ${\Gamma_\alpha[n]=\Gamma_{\alpha[n]}}$ )
{1{1{1{1`2^`}3}2^`}2{1`2^`}3} has recursion level ${\varphi(\Gamma_1,1)}$
{1{1{1{1`2^`}3}2^`}3{1`2^`}3} has recursion level ${\varphi(\Gamma_1,2)}$
{1{1{1{1`2^`}3}2^`}1`2{1`2^`}3} has recursion level ${\varphi(\Gamma_1+1,0)}$
{1{1{1{1`2^`}3}2^`}1{1{1{1`2^`}3}2^`}2{1`2^`}3} has recursion level ${\varphi(\Gamma_12,0)}$
{1{2{1{1`2^`}3}2^`}2{1`2^`}3} has recursion level ${\varphi(\omega^{\Gamma_1+1},0)}$
{1{1{1{1`2^`}3}3^`}2{1`2^`}3} has recursion level ${\varphi(\omega^{\Gamma_12},0)}$
{1{1{2{1`2^`}3}2^`}2{1`2^`}3} has recursion level ${\varphi(\omega^{\omega^{\omega^{\Gamma_1+1}}},0)}$
{1{1{1`2{1`2^`}3}2^`}2{1`2^`}3} has recursion level ${\varphi(\varepsilon_{\Gamma_1+1},0)}$
{1{1{1{1{1{1`2^`}3}2^`}2{1`2^`}3}2^`}2{1`2^`}3} has recursion level ${\varphi(\varphi(\Gamma_1,1),0)}$
{1{1{1{1{1{1{1`2^`}3}2^`}2{1{1`2^`}3}2^`}2{1`2^`}3}2^`}2{1`2^`}3} has recursion level ${\varphi(\varphi(\varphi(\Gamma_1,1),0),0)}$
{1{1`2^`}4} has recursion level ${\Gamma_2}$
{1{1`2^`}1,2} has recursion level ${\Gamma_\omega}$
{1{1`2^`}1.2} has recursion level ${\Gamma_{\varepsilon_0}}$
{1{1`2^`}1{1{2^`}2}2} has recursion level ${\Gamma_{\varphi(\omega,0)}}$
{1{1`2^`}1{1{1.2^`}2}2} has recursion level ${\Gamma_{\varphi(\varepsilon_0,0)}}$
{1{1`2^`}1{1{1`2^`}2}2} has recursion level ${\Gamma_{\Gamma_0}}$
{1{1`2^`}2{1{1`2^`}2}2} has recursion level ${\Gamma_{\Gamma_0+1}}$
{1{1`2^`}1{1{1`2^`}2}3} has recursion level ${\Gamma_{\Gamma_02}}$
{1{1`2^`}1{2{1`2^`}2}2} has recursion level ${\Gamma_{\omega^{\omega^{\Gamma_0+1}}}}$
{1{1`2^`}1{1{1`2^`}3}2} has recursion level ${\Gamma_{\Gamma_1}}$
{1{1`2^`}1{1{1`2^`}4}2} has recursion level ${\Gamma_{\Gamma_2}}$
{1{1`2^`}1{1{1`2^`}1{1{1`2^`}2}2}2} has recursion level ${\Gamma_{\Gamma_{\Gamma_0}}}$
{1{1`2^`}1{1{1`2^`}1{1{1`2^`}1{1{1`2^`}2}2}2}2} has recursion level ${\Gamma_{\Gamma_{\Gamma_{\Gamma_0}}}}$

Now φ function is not enough for the comparison. So we need to define fundamental sequences for θ function, and use them in FGH for comparisons. It’s difficult to define it. But there’s one solution: using fundamental sequences of Taranovsky’s notation directly.

{1{1`2^`}1`2} has recursion level ${\theta(\Omega+1,0)}$
{1{1`2^`}2`2} has recursion level ${\Gamma_{\theta(\Omega+1,0)+1}}$
{1{1`2^`}1{1{1`2^`}1`2}2`2} has recursion level ${\Gamma_{\theta(\Omega+1,0)2}}$
{1{1`2^`}1`3} has recursion level ${\theta(\Omega+1,1)}$
{1{1`2^`}1`1`2} has recursion level ${\theta(\Omega+2,0)}$
{1{1`2^`}1{2^`}2} has recursion level ${\theta(\Omega+\omega,0)}$
{1{1`2^`}1{1,2^`}2} has recursion level ${\theta(\Omega+\omega^\omega,0)}$
{1{1`2^`}1{1.2^`}2} has recursion level ${\theta(\Omega+\varepsilon_0,0)}$
{1{1`2^`}1{1{1{1`2^`}2}2^`}2} has recursion level ${\theta(\Omega+\Gamma_0,0)}$
{1{1`2^`}1{1{1{1`2^`}3}2^`}2} has recursion level ${\theta(\Omega+\Gamma_1,0)}$
{1{1`2^`}1{1{1{1`2^`}1`2}2^`}2} has recursion level ${\theta(\Omega+\theta(\Omega+1,0),0)}$
{1{1`2^`}1{1{1{1`2^`}1{1,2^`}2}2^`}2} has recursion level ${\theta(\Omega+\theta(\Omega+\omega^\omega,0),0)}$
{1{1`2^`}1{1{1{1`2^`}1{1{1{1`2^`}2}2^`}2}2^`}2} has recursion level ${\theta(\Omega+\theta(\Omega+\Gamma_0,0),0)}$
{1{1`2^`}1{1{1{1`2^`}1{1{1{1`2^`}1{1,2^`}2}2^`}2}2^`}2} has recursion level ${\theta(\Omega+\theta(\Omega+\theta(\Omega+\omega^\omega,0),0),0)}$
{1{1`2^`}1{1{1{1`2^`}1{1{1{1`2^`}1{1{1{1`2^`}1{1,2^`}2}2^`}2}2^`}2}2^`}2} has recursion level ${\theta(\Omega+\theta(\Omega+\theta(\Omega+\theta(\Omega+\omega^\omega,0),0),0),0)}$
{1{1`2^`}1{1`2^`}2} has recursion level ${\theta(\Omega2,0)}$
{1{1`2^`}2{1`2^`}2} has recursion level ${\theta(\Omega,\theta(\Omega2,0)+1)}$
{1{1`2^`}1`2{1`2^`}2} has recursion level ${\theta(\Omega+1,\theta(\Omega2,0)+1)}$
{1{1`2^`}1{1,2^`}2{1`2^`}2} has recursion level ${\theta(\Omega+\omega^\omega,\theta(\Omega2,0)+1)}$
{1{1`2^`}1{1{1{1`2^`}1{1`2^`}2}2^`}2{1`2^`}2} has recursion level ${\theta(\Omega+\theta(\Omega2,0),1)}$
{1{1`2^`}1{1{1{1`2^`}1{1`2^`}2}2^`}3{1`2^`}2} has recursion level ${\theta(\Omega+\theta(\Omega2,0),2)}$
{1{1`2^`}1{1{1{1`2^`}1{1`2^`}2}2^`}1`2{1`2^`}2} has recursion level ${\theta(\Omega+\theta(\Omega2,0)+1,0)}$
{1{1`2^`}1{1{1{1`2^`}1{1`2^`}2}2^`}1{1{1{1`2^`}1{1`2^`}2}2^`}2{1`2^`}2} has recursion level ${\theta(\Omega+\theta(\Omega2,0)2,0)}$
{1{1`2^`}1{2{1{1`2^`}1{1`2^`}2}2^`}2{1`2^`}2} has recursion level ${\theta(\Omega+\omega^{\theta(\Omega2,0)+1},0)}$
{1{1`2^`}1{1{1{1`2^`}1{1`2^`}2}3^`}2{1`2^`}2} has recursion level ${\theta(\Omega+\omega^{\theta(\Omega2,0)2},0)}$
{1{1`2^`}1{1{2{1`2^`}1{1`2^`}2}2^`}2{1`2^`}2} has recursion level ${\theta(\Omega+\omega^{\omega^{\theta(\Omega2,0)+1}},0)}$
{1{1`2^`}1{1{1`2{1`2^`}1{1`2^`}2}2^`}2{1`2^`}2} has recursion level ${\theta(\Omega+\varepsilon_{\theta(\Omega2,0)+1},0)}$
{1{1`2^`}1{1{1{1`2^`}2{1`2^`}2}2^`}2{1`2^`}2} has recursion level ${\theta(\Omega+\theta(\Omega,\theta(\Omega2,0)+1),0)}$
{1{1`2^`}1{1{1{1`2^`}1{1{1{1`2^`}1{1`2^`}2}2^`}2{1`2^`}2}2^`}2{1`2^`}2} has recursion level ${\theta(\Omega+\theta(\Omega+\theta(\Omega2,0),1),0)}$
{1{1`2^`}1{1{1{1`2^`}1{1{1{1`2^`}1{1{1{1`2^`}1{1`2^`}2}2^`}2{1`2^`}2}2^`}2{1`2^`}2}2^`}2{1`2^`}2} has recursion level ${\theta(\Omega+\theta(\Omega+\theta(\Omega+\theta(\Omega2,0),1),0),0)}$
{1{1`2^`}1{1`2^`}3} has recursion level ${\theta(\Omega2,1)}$
{1{1`2^`}1{1`2^`}1`2} has recursion level ${\theta(\Omega2+1,0)}$
{1{1`2^`}1{1`2^`}1{1`2^`}2} has recursion level ${\theta(\Omega3,0)}$
{1{2`2^`}2} has recursion level ${\theta(\Omega\omega,0)}$ , so s(n,n{1{2`2^`}2}2) reaches the limit of Extended Cascading-E Notation (i.e. En#{n}#n)
{1{2`2^`}3} has recursion level ${\theta(\Omega\omega,1)}$
{1{2`2^`}1`2} has recursion level ${\theta(\Omega\omega+1,0)}$
{1{2`2^`}1{2`2^`}2} has recursion level ${\theta(\Omega\omega2,0)}$
{1{3`2^`}2} has recursion level ${\theta(\Omega\omega^2,0)}$
{1{1,2`2^`}2} has recursion level ${\theta(\Omega\omega^\omega,0)}$
{1{1{1{1`2^`}2}2`2^`}2} has recursion level ${\theta(\Omega\Gamma_0,0)}$
{1{1{1{2`2^`}2}2`2^`}2} has recursion level ${\theta(\Omega\theta(\Omega\omega,0),0)}$
{1{1{1{1{1{1`2^`}2}2`2^`}2}2`2^`}2} has recursion level ${\theta(\Omega\theta(\Omega\Gamma_0,0),0)}$
{1{1`3^`}2} has recursion level ${\theta(\Omega^2,0)}$
{1{1`3^`}3} has recursion level ${\theta(\Omega^2,1)}$
{1{1`3^`}1`2} has recursion level ${\theta(\Omega^2+1,0)}$
{1{1`3^`}1{1`3^`}2} has recursion level ${\theta(\Omega^22,0)}$
{1{2`3^`}2} has recursion level ${\theta(\Omega^2\omega,0)}$
{1{1,2`3^`}2} has recursion level ${\theta(\Omega^2\omega^\omega,0)}$
{1{1{1{1`3^`}2}2`3^`}2} has recursion level ${\theta(\Omega^2\theta(\Omega^2,0),0)}$
{1{1`4^`}2} has recursion level ${\theta(\Omega^3,0)}$
{1{1`5^`}2} has recursion level ${\theta(\Omega^4,0)}$
{1{1`1,2^`}2} has recursion level ${\theta(\Omega^\omega,0)}$ , so s(n,n{1{1`1,2^`}2}2) is comparable to weak tree function (i.e. tree(n))
{1{2`1,2^`}2} has recursion level ${\theta(\Omega^\omega\omega,0)}$ , so s(n,n{1{2`1,2^`}2}2) is the best known lower bound for the growth rate of TREE function
{1{1`1{1{1`2^`}2}2^`}2} has recursion level ${\theta(\Omega^{\Gamma_0},0)}$
{1{1`1{1{1`3^`}2}2^`}2} has recursion level ${\theta(\Omega^{\theta(\Omega^2,0)},0)}$
{1{1`1{1{1`1{1{1`3^`}2}2^`}2}2^`}2} has recursion level ${\theta(\Omega^{\theta(\Omega^{\theta(\Omega^2,0)},0)},0)}$
{1{1`1`2^`}2} has recursion level ${\theta(\Omega^\Omega,0)}$ , so s(n,n{1{1`1`2^`}2}2) reaches the limit of dimensional array notation of Hyperfactorial array notation (i.e. n![1([1(…[1([1])2]…)2])2] with n nests of []’s)
{1{1`1{1{1`1`2^`}2}2^`}2{1`1`2^`}2} has recursion level ${\theta(\Omega^{\theta(\Omega^\Omega,0)},1)}$
{1{1`1`2^`}3} has recursion level ${\theta(\Omega^\Omega,1)}$
{1{1`1`2^`}1`2} has recursion level ${\theta(\Omega^\Omega+1,0)}$
{1{1`1`2^`}1{1`1`2^`}2} has recursion level ${\theta(\Omega^\Omega2,0)}$
{1{2`1`2^`}2} has recursion level ${\theta(\Omega^\Omega\omega,0)}$
{1{1{1{1`1`2^`}2}2`1`2^`}2} has recursion level ${\theta(\Omega^\Omega\theta(\Omega^\Omega,0),0)}$
{1{1`2`2^`}2} has recursion level ${\theta(\Omega^{\Omega+1},0)}$
{1{1`1,2`2^`}2} has recursion level ${\theta(\Omega^{\Omega+\omega},0)}$
{1{1`1{1{1`1`2^`}2}2`2^`}2} has recursion level ${\theta(\Omega^{\Omega+\theta(\Omega^\Omega,0)},0)}$
{1{1`1`3^`}2} has recursion level ${\theta(\Omega^{\Omega2},0)}$
{1{1`1`3^`}3} has recursion level ${\theta(\Omega^{\Omega2},1)}$
{1{2`1`3^`}2} has recursion level ${\theta(\Omega^{\Omega2}\omega,0)}$
{1{1`2`3^`}2} has recursion level ${\theta(\Omega^{\Omega2+1},0)}$
{1{1`1`4^`}2} has recursion level ${\theta(\Omega^{\Omega3},0)}$
{1{1`1`1,2^`}2} has recursion level ${\theta(\Omega^{\Omega\omega},0)}$
{1{1`1`1`2^`}2} has recursion level ${\theta(\Omega^{\Omega^2},0)}$
{1{1`2`1`2^`}2} has recursion level ${\theta(\Omega^{\Omega^2+1},0)}$
{1{1`1`2`2^`}2} has recursion level ${\theta(\Omega^{\Omega^2+\Omega},0)}$
{1{1`1`1`3^`}2} has recursion level ${\theta(\Omega^{\Omega^22},0)}$
{1{1`1`1`1`2^`}2} has recursion level ${\theta(\Omega^{\Omega^3},0)}$
{1{1{2^`}2^`}2} has recursion level ${\theta(\Omega^{\Omega^\omega},0)}$
{1{1`2{2^`}2^`}2} has recursion level ${\theta(\Omega^{\Omega^\omega+1},0)}$
{1{1`1`2{2^`}2^`}2} has recursion level ${\theta(\Omega^{\Omega^\omega+\Omega},0)}$
{1{1`1`1`2{2^`}2^`}2} has recursion level ${\theta(\Omega^{\Omega^\omega+\Omega^2},0)}$
{1{1{2^`}3^`}2} has recursion level ${\theta(\Omega^{\Omega^\omega2},0)}$
{1{1{2^`}1`2^`}2} has recursion level ${\theta(\Omega^{\Omega^{\omega+1}},0)}$
{1{1{2^`}1{2^`}2^`}2} has recursion level ${\theta(\Omega^{\Omega^{\omega2}},0)}$
{1{1{3^`}2^`}2} has recursion level ${\theta(\Omega^{\Omega^{\omega^2}},0)}$
{1{1{1{1{1{2^`}2^`}2}2^`}2^`}2} has recursion level ${\theta(\Omega^{\Omega^{\theta(\Omega^{\Omega^\omega},0)}},0)}$
{1{1{1`2^`}2^`}2} has recursion level ${\theta(\Omega^{\Omega^\Omega},0)}$
{1{1{1`2^`}2^`}3} has recursion level ${\theta(\Omega^{\Omega^\Omega},1)}$
{1{1{1`2^`}2^`}1`2} has recursion level ${\theta(\Omega^{\Omega^\Omega}+1,0)}$
{1{2{1`2^`}2^`}2} has recursion level ${\theta(\Omega^{\Omega^\Omega}\omega,0)}$
{1{1`2{1`2^`}2^`}2} has recursion level ${\theta(\Omega^{\Omega^\Omega+1},0)}$
{1{1{1`2^`}3^`}2} has recursion level ${\theta(\Omega^{\Omega^\Omega2},0)}$
{1{1{1`2^`}1`2^`}2} has recursion level ${\theta(\Omega^{\Omega^{\Omega+1}},0)}$
{1{1{1`2^`}1{1`2^`}2^`}2} has recursion level ${\theta(\Omega^{\Omega^{\Omega2}},0)}$
{1{1{2`2^`}2^`}2} has recursion level ${\theta(\Omega^{\Omega^{\Omega\omega}},0)}$
{1{1{1`3^`}2^`}2} has recursion level ${\theta(\Omega^{\Omega^{\Omega^2}},0)}$
{1{1{1`1{1{1{1`2^`}2^`}2}2^`}2^`}2} has recursion level ${\theta(\Omega^{\Omega^{\Omega^{\theta(\Omega^{\Omega^\Omega},0)}}},0)}$
{1{1{1`1`2^`}2^`}2} has recursion level ${\theta(\Omega^{\Omega^{\Omega^\Omega}},0)}$
{1{1{1{2^`}2^`}2^`}2} has recursion level ${\theta(\Omega^{\Omega^{\Omega^{\Omega^\omega}}},0)}$
{1{1{1{1`2^`}2^`}2^`}2} has recursion level ${\theta(\Omega^{\Omega^{\Omega^{\Omega^\Omega}}},0)}$
{1{1{1{1`1`2^`}2^`}2^`}2} has recursion level ${\theta(\Omega^{\Omega^{\Omega^{\Omega^{\Omega^\Omega}}}},0)}$
{1{1{1{1{1`1`2^`}2^`}2^`}2^`}2} has recursion level ${\theta(\Omega^{\Omega^{\Omega^{\Omega^{\Omega^{\Omega^{\Omega^\Omega}}}}}},0)}$

Let f(n) = s(n,n{1{1{1…{1{1,2^`}2^`}…2^`}2^`}2}2) where there’re n separators with a grave accent. So f(n) is the limit of EAN. It eventually outgrows all above, and has growth rate ${\theta(\varepsilon_{\Omega+1},0)}$ .

10 thoughts on “Expanding array notation”

Samuel Fields says:

Hypcos how would you describe Sbiis Saibian’s number Monster-Giant with your (EAN)?

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2016-09-19 at 02:39
- hypcos says:
  
  It’s approximate s(100,100{1{1`2}1,2`2}2).
  
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  2016-09-19 at 07:01
SuperSpruce says:

I’m finally beginning to understand the dot separators! Yay!

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2019-01-01 at 06:16
SuperSpruce says:

Where does the A_t separators come into play after step 3 of case B2? Are the A_t separators and B_t separators somehow related in some way?

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2019-03-11 at 03:11
- hypcos says:
  
  Before case B2, the process may run case B3 many times, each time modifying the array. B_t is a part of the array, so it may be modified by case B3. A_t is the unmodified copy of B_t created by case B3.
  
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  2019-03-11 at 13:03
  - SuperSpruce says:
    
    And is the maximal t we find from A_t the same as the t we use for B_t in the next couple steps?
    
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    2019-04-29 at 02:10
    - hypcos says:
      
      Yes
      
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      2019-05-05 at 07:29
      - Dessa Hall says:
        
        Limit of EAN is Lugubrigoth (approximately {100, 100 [1 [1 ¬ 3] 2] 2}
        
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        2023-09-02 at 07:15
Scorcher says:

Why ε_0+1 is (n,n(2’2)2)?
Why not (n,n,2(1’2)2)?
Is it (n,n,2(1’2)2) = (n,(n,(n,…(1’2)2)(1’2)2)(1’2)2) – n-times?
Is FS for FGH_ε_0+1(n) = FGH_ε_0(FGH_ε_0(FGH_ε_0(…n…))) – n-times?

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2020-10-14 at 21:39
Dessa Hall says:

s(n,n{1{1`2^`}2}2) is gamma-zero?
s(10,100{1{1`2^`}2}2) = Kungulus
s(n,n{1{1`2^`}2}2) is gamma-zero. Correction.

≈ s(100,100{1{1`2^`}2}2) – Pentacthulhum.

≈ s(100,100{1{1`2^`}1{1`2^`}2}2) – Hexacthulhum.

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2023-09-02 at 07:20