Expanding array notation

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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 remains 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 “Sb 2 {1.k} n-1″, where b is the iterator, S1 is comma, and Si+1 = {1 Si 2.k-1} (k > 1) or Si+1 = {1 Si 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 Sb, where b is the iterator, S1 = “{n-1 #}” and Si+1 = “Si 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) = {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 Sb, where b is the iterator, S1 is comma and Si+1 = {1`…1`1 Si 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 b, n, Bt and K are parts of the original array but t, i, Si, At, P and Q 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 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 Bt 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 maximum of t such that At hasn’t a grave accent at the left-superscript position of the rbrace.
      4. Let string P and Q be such that Bt = “P ` n Q”
      5. Change Bt into Sb, where b is the iterator, S1 is comma, and Si+1 = “P 1 Si 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 At = 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 Sb, where b is the iterator, S1 = “{n-1 #}” and Si+1 = “Si 1 {n-1 #}”.
      2. The process ends.

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 At′s. But when we meet case B2 after all the pulling-out, the separators surrounding the grave accent are Bt′s (and the t is how many layer the At′s and Bt′s are at). In step 3 of case B2, the At 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 Bt expands into b nests, replacing the grave accent with one less nests of Bt 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 At 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)}.

2 thoughts on “Expanding array notation

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