Extended Expanding Array Notation
Introduction
Like with the previous chapter, we can apply the similar rules to the Primitive Expanding Array Notation right now.
Rules
Reuse the previous rules (including similar rules).
In this part, as we introduced \ (backslash). The \ symbol is also called "ranking separator". It works similarly to the ,, in strong array notation (SAN).
When you have a backslash, it works like this: First find the "diagonalization part" of the separator with the \. To do this, look for the first part separated by \'s that doesn't start with 1 (if it just starts with 1, decompose it normally). Then replace the last n by n-1, and the previous 1 becomes the "nesting point" - that's where the layers are repeated.
So 1{1 \ 2}2 becomes nests of 1{1{x \ 1}2}2, while 1{1{1 \ 2}3}2 becomes nests of 1{1{x}2{1 \ 2}2}2. 1{1 \ 1 \ 2}3 becomes nests of 1{1 \ 1{1 \ x \ 2}2}2{1 \ 1 \ 2}2, and so on.
In fact, {x \ 1} is usually equal {x}, {x \ 2} is normally <x>, {x \ 3} is normally <<x>>, {x \ y} is normally <<<...<<<x>>>...>>> with y-1 pairs of angle brackets, and so on. And {0 \ a} corresponds to Ω_(a-1) for 1 ≤ a < 1,2, but Ω_a for a ≥ 1,2. This is not always the case, however.
Previous level examples
1{1 \ 2}2 = 1{1 / 2}2
1{2 \ 2}2 = 1{2 / 2}2
1{1 {1 {1 \ 2} 2} 2 \ 2} = 1{1{1 / 2}2 / 2}2
1{1 {1 \ 2} 2 \ 2} = 1{1 / 3}2
1{1 {1 {1 \ 2} 3} 1 {1 \ 2} 2 \ 2} = 1{1{1 / 3}2 / 3}2
1{1 {1 \ 2} 3 \ 2}2 = 1{1 / 4}2
1{1 {1 \ 2} 1,2 \ 2}2 = 1{1 / 1,2}2
1{1 {1 \ 2} 1 {1 {1 \ 2} 2} 2 \ 2}2 = 1{1 / 1{1 / 2}2}2
1{1 {1 \ 2} 1 {1 \ 2} 2 \ 2}2 = 1{1 / 1 / 2}2
1{1 {1 \ 2} 1 {1 \ 2} 1 {1 \ 2} 2 \ 2}2 = 1{1 / 1 / 1 / 2}2
1{1 {2 \ 2} 2 \ 2}2 = 1{1 <2> 2}2
1{1 {1 {1 {1 \ 2} 2} 2 \ 2} 2 \ 2}2 = 1{1 <1{1 / 2}2> 2}2
1{1 {1 {1 \ 2} 2 \ 2} 2 \ 2}2 = 1{1 <1 / 2> 2}2
1{1 {1 {1 \ 2} 1 {1 \ 2} 2 \ 2} 2 \ 2}2 = 1{1 <1 / 1 / 2> 2}2
1{1 {1 {1 {1 \ 2} 2 \ 2} 2 \ 2} 2 \ 2}2 = 1{1 <1 <1 / 2> 2> 2}2
1{1 \ 3}2 = 1{1 // 2}2
1{1 \ 4}2 = 1{1 /// 2}2
1{1 \ 5}2 = 1{1 //// 2}2
Analysis
/// WIP ///
Here're comparisons between my array notation and FGH. If separator A has recursion level α, then a[A]b has growth rate ω^ω^α.
OCF extension credited to: https://googology.fandom.com/wiki/User:DeepLineMadom/DeepLineMadom%27s_extension_of_the_extended_Buchholz%27s_function
1{1 \ 1 \ 2}2 has level ψ(I) = ψ(Φ(1,0)) = ψ(M)
1{2 \ 1 \ 2}2 has level ψ(I)^ω = ω^ω^(ψ(Φ(1,0))+1)
1{1 {1 {1 \ 1 \ 2} 2} 2}2 \ 1 \ 2}2 has level ψ(I)^ψ(I) = ω^ω^(ψ(Φ(1,0))*2)
1{1 {1 \ 2} 2 \ 1 \ 2}2 has level ψ(I+Ω) = ε(ψ(Φ(1,0))+1)
1{1 {1 \ 2} 1 {1 \ 2} 2 \ 1 \ 2}2 has level ψ(I+Ω^2) = ζ(ψ(Φ(1,0))+1)
1{1 {1 {1 \ 2} 2 \ 2} 2 \ 1 \ 2}2 has level ψ(I+Ω^Ω) = Γ(ψ(Φ(1,0))+1)
1{1 {1 {1 \ 3} 2 \ 2} 2 \ 1 \ 2}2 has level ψ(I+ψ_1(Ω_2)) = ψ(Φ(1,0)+ε(Ω+1))
1{1 {1 {1 \ 1,2} 2 \ 2} 2 \ 1 \ 2}2 has level ψ(I+ψ_1(Ω_ω)) = ψ(Φ(1,0)+ψ_1(Ω_ω))
1{1 {1 {1 \ 1 {1 \ 2} 2} 2 \ 2} 2 \ 1 \ 2}2 has level ψ(I+ψ_1(Ω_Ω)) = ψ(Φ(1,0)+ψ_1(Ω_Ω))
1{1 {1 {1 \ 1 \ 2} 2 \ 2} 2 \ 1 \ 2}2 has level ψ(I+ψ_1(I)) = ψ(Φ(1,0)+ψ_1(Φ(1,0)))
1{1 {1 \ 3} 2 \ 1 \ 2}2 has level ψ(I+Ω_2) = ψ(Φ(1,0)+Ω_2)
1{1 {1 {1 \ 1 \ 2} 2 \ 3} 2 \ 1 \ 2}2 has level ψ(I+ψ_2(I)) = ψ(Φ(1,0)+ψ_2(Φ(1,0)))
1{1 {1 \ 4} 2 \ 1 \ 2}2 has level ψ(I+Ω_2) = ψ(Φ(1,0)+Ω_3)
1{1 {1 \ 1,2} 2 \ 1 \ 2}2 has level ψ(I+Ω_ω) = ψ(Φ(1,0)+Ω_ω)
1{1 {1 \ 1 {1 \ 2} 2} 2 \ 1 \ 2}2 has level ψ(I+Ω_Ω) = ψ(Φ(1,0)+Ω_Ω)
1{1 {1 \ 1 {1 \ 1 \ 2} 2} 2 \ 1 \ 2}2 has level ψ(I+ψ_I(I)) = ψ(Φ(1,0)*2)
1{1 {1 \ 2 {1 \ 1 \ 2} 2} 2 \ 1 \ 2}2 has level ψ(I+ψ_I(I+1)) = ψ(ε(Φ(1,0)+1)) = ψ(Ω(Φ(1,0)+1))
1{1 {1 \ 1 {1 \ 1 {1 \ 1 \ 2} 2} 2 {1 \ 1 \ 2} 2} 2 \ 1 \ 2}2 has level ψ(I+ψ_I(I+ψ_I(I))) = ψ(Ω(Φ(1,0)2))
1{1 {1 \ 1 {1 \ 2 {1 \ 1 \ 2} 2} 2 {1 \ 1 \ 2} 2} 2 \ 1 \ 2}2 has level ψ(I+ψ_I(I+ψ_I(I+1))) = ψ(Ω(Ω(Φ(1,0)+1)))
1{1 {1 \ 1 \ 2} 2 \ 1 \ 2}2 has level ψ(I2) = ψ(Φ(1,1))
1{1 {1 \ 1 {1 \ 1 \ 2} 3 \ 2} 2 \ 1 \ 2}2 has level ψ(I2+ψ_I(I2)) = ψ(Φ(1,1)2)
1{1 {1 \ 2 {1 \ 1 \ 2} 3 \ 2} 2 \ 1 \ 2}2 has level ψ(I2+ψ_I(I2+1)) = ψ(Ω(Φ(1,1)+1))
1{1 {1 \ 1 {1 \ 1 {1 \ 1 \ 2} 3 \ 2} 3 \ 2} 2 \ 1 \ 2}2 has level ψ(I2+ψ_I(I2+ψ(I2))) = ψ(Ω(Φ(1,1)2))
1{1 {1 \ 1 \ 2} 3 \ 1 \ 2}2 has level ψ(I3) = ψ(Φ(1,2))
1{1 {1 \ 1 \ 2} 4 \ 1 \ 2}2 has level ψ(I4) = ψ(Φ(1,3))
1{1 {1 \ 1 \ 2} 1,2 \ 1 \ 2}2 has level ψ(Iω) = ψ(Φ(1,ω))
1{1 {1 \ 1 \ 2} 1 {1 {1 \ 1 \ 2} 2} 2 \ 1 \ 2}2 has level ψ(Iψ(I)) = ψ(Φ(1,ψ(Φ(1,0))))
1{1 {1 \ 1 \ 2} 1 {1 \ 2} 2 \ 1 \ 2}2 has level ψ(IΩ) = ψ(Φ(1,Ω))
1{1 {1 \ 1 \ 2} 2 {1 \ 2} 2 \ 1 \ 2}2 has level ψ(IΩ+I) = ψ(Φ(1,Ω+1))
1{1 {1 \ 1 \ 2} 1 {1 \ 2} 3 \ 1 \ 2}2 has level ψ(IΩ2) = ψ(Φ(1,Ω2))
1{1 {1 \ 1 \ 2} 1 {2 \ 2} 2 \ 1 \ 2}2 has level ψ(IΩ^ω) = ψ(Φ(1,Ω^ω))
1{1 {1 \ 1 \ 2} 1 {1 \ 3} 2 \ 1 \ 2}2 has level ψ(IΩ_2) = ψ(Φ(1,Ω_2))
1{1 {1 \ 1 \ 2} 1 {1 \ 1,2} 2 \ 1 \ 2}2 has level ψ(IΩ_ω) = ψ(Φ(1,Ω_ω))
1{1 {1 \ 1 \ 2} 1 {1 \ 1 {1 \ 2} 2} 2 \ 1 \ 2}2 has level ψ(IΩ_ω) = ψ(Φ(1,Ω_Ω))
1{1 {1 \ 1 \ 2} 1 {1 \ 1 {1 \ 1 \ 2} 2} 2 \ 1 \ 2}2 has level ψ(Iψ_I(I)) = ψ(Φ(1,Φ(1,0)))
1{1 {1 \ 1 \ 2} 1 {1 \ 2 {1 \ 1 \ 2} 2} 2 \ 1 \ 2}2 has level ψ(Iψ_I(I+1)) = ψ(Φ(1,Ω(Φ(1,0)+1)))
1{1 {1 \ 1 \ 2} 1 {1 \ 1 {1 \ 1 \ 2} 3} 2 \ 1 \ 2}2 has level ψ(Iψ_I(I2)) = ψ(Φ(1,Φ(1,1)))
1{1 {1 \ 1 \ 2} 1 {1 \ 1 {1 \ 1 \ 2} 1,2} 2 \ 1 \ 2}2 has level ψ(Iψ_I(Iω)) = ψ(Φ(1,Φ(1,ω)))
1{1 {1 \ 1 \ 2} 1 {1 \ 1 {1 \ 1 {1 \ 1 \ 2} 2} 2} 2 \ 1 \ 2}2 has level ψ(Iψ_I(Iψ_I(I))) = ψ(Φ(1,Φ(1,Φ(1,0))))
1{1 {1 \ 1 \ 2} 1 {1 \ 1 \ 2} 2 \ 1 \ 2}2 has level ψ(I^2) = ψ(Φ(2,0))
1{1 {1 \ 1 {1 \ 1 \ 2} 2} 2 {1 \ 1 \ 2} 1 {1 \ 1 \ 2} 2 \ 1 \ 2}2 has level ψ(I^2+ψ_I(I))
1{1 {1 \ 1 {1 \ 1 \ 2} 1 {1 \ 1 \ 2} 2} 2 {1 \ 1 \ 2} 1 {1 \ 1 \ 2} 2 \ 1 \ 2}2 has level ψ(I^2+ψ_I(I^2))
1{1 {1 \ 1 \ 2} 2 {1 \ 1 \ 2} 2 \ 1 \ 2}2 has level ψ(I^2+I) = ψ(Φ(1,Φ(2,0)+1))
1{1 {1 \ 1 \ 2} 1 {1 \ 1 {1 \ 1 \ 2} 1 {1 \ 1 \ 2} 2} 2 {1 \ 1 \ 2} 2 \ 1 \ 2}2 has level ψ(I^2+Iψ_I(I^2)) = ψ(Φ(1,Φ(2,0)2))
1{1 {1 \ 1 \ 2} 1 {1 \ 1 \ 2} 3 \ 1 \ 2}2 has level ψ(I^2*2) = ψ(Φ(2,1))
1{1 {1 \ 1 \ 2} 1 {1 \ 1 \ 2} 1,2 \ 1 \ 2}2 has level ψ(I^2*ω) = ψ(Φ(2,ω))
1{1 {1 \ 1 \ 2} 1 {1 \ 1 \ 2} 1 {1 \ 2} 2 \ 1 \ 2}2 has level ψ(I^2*Ω) = ψ(Φ(2,Ω))
1{1 {1 \ 1 \ 2} 1 {1 \ 1 \ 2} 1 {1 \ 1 {1 \ 1 \ 2} 2} 2 \ 1 \ 2}2 has level ψ(I^2*ψ_I(I^2)) = ψ(Φ(2,Φ(2,0)))
1{1 {1 \ 1 \ 2} 1 {1 \ 1 \ 2} 1 {1 \ 1 \ 2} 2 \ 1 \ 2}2 has level ψ(I^3) = ψ(Φ(3,0))
1{1 {1 \ 1 \ 2} 1 {1 \ 1 \ 2} 1 {1 \ 1 \ 2} 1 {1 \ 1 \ 2} 2 \ 1 \ 2}2 has level ψ(I^4) = ψ(Φ(4,0))
1{1 {2 \ 1 \ 2} 2 \ 1 \ 2}2 has level ψ(I^ω) = ψ(Φ(ω,0))
1{1 {1 {1 \ 2} 2 \ 1 \ 2} 2 \ 1 \ 2}2 has level ψ(I^Ω) = ψ(Φ(Ω,0))
1{1 {1 {1 \ 1 {1 \ 1 \ 2} 2} 2 \ 1 \ 2} 2 \ 1 \ 2}2 has level ψ(I^ψ_I(I)) = ψ(Φ(Φ(1,0),0))
1{1 {1 {1 \ 1 \ 2} 2 \ 1 \ 2} 2 \ 1 \ 2}2 has level ψ(I^I) = ψ(Φ(1,0,0))
1{1 {1 {1 \ 1 \ 2} 2 \ 1 \ 2} 3 \ 1 \ 2}2 has level ψ(I^I*2) = ψ(Φ(1,0,1))
1{1 {1 {1 \ 1 \ 2} 2 \ 1 \ 2} 1 {1 \ 1 \ 2} 2 \ 1 \ 2}2 has level ψ(I^(I+1)) = ψ(Φ(1,1,0))
1{1 {1 {1 \ 1 \ 2} 2 \ 1 \ 2} 1 {1 {1 \ 1 \ 2} 2 \ 1 \ 2} 2 \ 1 \ 2}2 has level ψ(I^(I2)) = ψ(Φ(2,0,0))
1{1 {2 {1 \ 1 \ 2} 2 \ 1 \ 2} 2 \ 1 \ 2}2 has level ψ(I^(Iω)) = ψ(Φ(ω,0,0))
1{1 {1 {1 \ 1 \ 2} 3 \ 1 \ 2} 2 \ 1 \ 2}2 has level ψ(I^I^2) = ψ(Φ(1,0,0,0))
1{1 {1 {1 \ 1 \ 2} 1,2 \ 1 \ 2} 2 \ 1 \ 2}2 has level ψ(I^I^ω)
1{1 {1 {1 \ 1 \ 2} 1 {1 \ 1 \ 2} 2 \ 1 \ 2} 2 \ 1 \ 2}2 has level ψ(I^I^I)
1{1 {1 {1 {1 \ 1 \ 2} 2 \ 1 \ 2} 2 \ 1 \ 2} 2 \ 1 \ 2}2 has level ψ(I^I^I^I)
Moving on...
1{1 \ 2 \ 2}2 has level ψ(Ω(I+1)) = ψ(ε(I+1)) = ψ(M+ψ_M(M+1))
1{1 {1 \ 2} 2 \ 2 \ 2}2 has level ψ(Ω(I+1)+Ω) = ε(ψ(ε(I+1))+1)
1{1 {1 \ 1,2} 2 \ 2 \ 2} 2 has level ψ(Ω(I+1)+Ω_ω)
1{1 {1 \ 1 {1 \ 1 \ 2} 2} 2 \ 2 \ 2}2 has level ψ(Ω(I+1)+ψ_I(I))
1{1 {1 \ 1 \ 2} 2 \ 2 \ 2}2 has level ψ(Ω(I+1)+I)
1{1 {1 \ 1 \ 2} 1 {1 \ 1 \ 2} 2 \ 2 \ 2}2 has level ψ(Ω(I+1)+I^2)
1{1 {2 \ 1 \ 2} 2 \ 2 \ 2}2 has level ψ(Ω(I+1)+I^ω)
1{1 {1 {1 \ 1 \ 2} 2 \ 1 \ 2} 2 \ 2 \ 2}2 has level ψ(Ω(I+1)+I^I)
1{1 {1 {1 \ 2 \ 2} 2 \ 1 \ 2} 2 \ 2 \ 2}2 has level ψ(Ω(I+1)+ψ_(I+1)(Ω(I+1)))
1{1 {1 {1 \ 2 \ 2} 2 \ 1 \ 2} 3 \ 2 \ 2}2 has level ψ(Ω(I+1)+ψ_(I+1)(Ω(I+1))2)
1{1 {1 {1 \ 2 \ 2} 2 \ 1 \ 2} 1 {1 {1 \ 2 \ 2} 2 \ 1 \ 2} 2 \ 2 \ 2}2 has level ψ(Ω(I+1)+ψ_(I+1)(Ω(I+1))^2)
1{1 {2 {1 \ 2 \ 2} 2 \ 1 \ 2} 2 \ 2 \ 2}2 has level ψ(Ω(I+1)+ψ_(I+1)(Ω(I+1))^ω)
1{1 {1 {1 {1 \ 2 \ 2} 2 \ 1 \ 2} 2 {1 \ 2 \ 2} 2 \ 1 \ 2} 2 \ 2 \ 2}2 has level ψ(Ω(I+1)+ψ_(I+1)(Ω(I+1))^ψ_(I+1)(Ω(I+1)))
1{1 {1 \ 2 \ 2} 2 \ 2 \ 2}2 has level ψ(Ω(I+1)2)
1{1 {1 \ 2 \ 2} 3 \ 2 \ 2}2 has level ψ(Ω(I+1)3)
1{1 {1 \ 2 \ 2} 1,2 \ 2 \ 2}2 has level ψ(Ω(I+1)ω)
1{1 {1 \ 2 \ 2} 1 {1 \ 2} 2 \ 2 \ 2}2 has level ψ(Ω(I+1)Ω)
1{1 {1 \ 2 \ 2} 1 {1 \ 1 \ 2} 2 \ 2 \ 2}2 has level ψ(Ω(I+1)I)
1{1 {1 \ 2 \ 2} 1 {1 \ 1 \ 2} 3 \ 2 \ 2}2 has level ψ(Ω(I+1)I2)
1{1 {1 \ 2 \ 2} 1 {1 \ 1 \ 2} 1 {1 \ 1 \ 2} 2 \ 2 \ 2}2 has level ψ(Ω(I+1)I^2)
1{1 {1 \ 2 \ 2} 1 {1 {1 \ 1 \ 2} 2 \ 1 \ 2} 2 \ 2 \ 2}2 has level ψ(Ω(I+1)I^2)
1{1 {1 \ 2 \ 2} 1 {1 \ 2 \ 2} 2 \ 2 \ 2}2 has level ψ(Ω(I+1)^2)
1{1 {1 \ 2 \ 2} 1 {1 \ 2 \ 2} 1 {1 \ 2 \ 2} 2 \ 2 \ 2}2 has level ψ(Ω(I+1)^3)
1{1 {2 \ 2 \ 2} 2 \ 2 \ 2}2 has level ψ(Ω(I+1)^ω)
1{1 {1 {1 \ 1 \ 2} 2 \ 2 \ 2} 2 \ 2 \ 2}2 has level ψ(Ω(I+1)^I)
1{1 {1 {1 \ 2 \ 2} 2 \ 2 \ 2} 2 \ 2 \ 2}2 has level ψ(Ω(I+1)^Ω(I+1))
1{1 {1 {1 \ 2 \ 2} 1 {1 \ 2 \ 2} 2 \ 2 \ 2} 2 \ 2 \ 2}2 has level ψ(Ω(I+1)^Ω(I+1)^Ω(I+1))
1{1 \ 3 \ 2}2 has level ψ(Ω(I+2)) = ψ(ε(Ω(I+1)+1)) = ψ(M+ψ_M(M+2))
1{1 {1 \ 3 \ 2} 2 \ 3 \ 2}2 has level ψ(Ω(I+2)2)
1{1 \ 4 \ 2}2 has level ψ(Ω(I+3)) = ψ(ε(Ω(I+2)+1)) = ψ(M+ψ_M(M+3))
1{1 \ 1,2 \ 2}2 has level ψ(Ω(I+ω))
1{1 \ 1 {1 \ 2} 2 \ 2}2 has level ψ(Ω(I+Ω))
1{1 \ 1 {1 \ 1 {1 \ 1 \ 2} 2} 2 \ 2}2 has level ψ(Ω(I+ψ_I(I)))
1{1 \ 1 {1 \ 1 \ 2} 2 \ 2}2 has level ψ(Ω(I2)) = ψ(M+ψ_M(M+ψ_M(M)))
1{1 \ 2 {1 \ 1 \ 2} 2 \ 2}2 has level ψ(Ω(I2+1))
1{1 \ 1 {1 \ 1 \ 2} 3 \ 2}2 has level ψ(Ω(I3))
1{1 \ 1 {1 \ 1 \ 2} 1 {1 \ 1 \ 2} 2 \ 2}2 has level ψ(Ω(I^2))
1{1 \ 1 {2 \ 1 \ 2} 2 \ 2}2 has level ψ(Ω(I^ω))
1{1 \ 1 {1 \ 2 \ 2} 2 \ 2}2 has level ψ(Ω(Ω(I+1))) = ψ(M+ψ_M(M+ψ_M(M+1)))
1{1 \ 1 {1 \ 1 {1 \ 1 \ 2} 2 \ 2} 2 \ 2}2 has level ψ(Ω(Ω(I2))) = ψ(M+ψ_M(M+ψ_M(M+ψ_M(M))))
Then...
1{1 \ 1 \ 3}2 has level ψ(I_2) = ψ(M2)
1{1 {1 \ 1 \ 2} 2 \ 1 \ 3}2 has level ψ(I_2+I)
1{1 {1 \ 2 \ 2} 2 \ 1 \ 3}2 has level ψ(I_2+Ω(I+1))
1{1 {1 \ 1 {1 \ 1 \ 2} 2 \ 2} 2 \ 1 \ 3}2 has level ψ(I_2+Ω(I2))
1{1 {1 \ 1 {1 \ 1 \ 3} 2 \ 2} 2 \ 1 \ 3}2 has level ψ(I_2+ψ_{I_2}(I2))
1{1 {1 \ 1 \ 3} 2 \ 1 \ 3}2 has level ψ(I_2*2)
1{1 {1 \ 1 \ 3} 1,2 \ 1 \ 3}2 has level ψ(I_2*ω)
1{1 {1 \ 1 \ 3} 1 {1 \ 2} 2 \ 1 \ 3}2 has level ψ(I_2*Ω)
1{1 {1 \ 1 \ 3} 1 {1 \ 1 \ 2} 2 \ 1 \ 3}2 has level ψ(I_2*I)
1{1 {1 \ 1 \ 3} 1 {1 \ 1 {1 \ 1 \ 3} 2 \ 2} 2} 2 \ 1 \ 3}2 has level ψ(I_2*ψ_{I_2}(I2))
1{1 {1 \ 1 \ 3} 1 {1 \ 1 \ 3} 2 \ 1 \ 3}2 has level ψ(I_2^2)
1{1 {1 \ 1 \ 3} 1 {1 \ 1 \ 3} 1 {1 \ 1 \ 3} 2 \ 1 \ 3}2 has level ψ(I_2^3)
1{1 {2 \ 1 \ 3} 2 \ 1 \ 3}2 has level ψ(I_2^ω)
1{1 {1 {1 \ 1 \ 3} 2 \ 1 \ 3} 2 \ 1 \ 3}2 has level ψ(I_2^I_2)
1{1 \ 2 \ 3}2 has level ψ(Ω(I_2+1)) = ψ(M2+ψ_M(M2+1))
1{1 \ 1,2 \ 3}2 has level ψ(Ω(I_2+ω))
1{1 \ 1 {1 \ 1 \ 2} 2 \ 3}2 has level ψ(Ω(I_2+I)) = ψ(M2+ψ_M(M2+ψ_M(M)))
1{1 \ 1 {1 \ 1 \ 3} 2 \ 3}2 has level ψ(Ω(I_2*2)) = ψ(M2+ψ_M(M2+ψ_M(M2)))
1{1 \ 1 {1 \ 2 \ 3} 2 \ 3}2 has level ψ(Ω(Ω(I_2+1))) = ψ(M2+ψ_M(M2+ψ_M(M2+1)))
1{1 \ 1 \ 4}2 has level ψ(I_3) = ψ(M3)
1{1 \ 2 \ 4}2 has level ψ(Ω(I_3+1)) = ψ(M3+ψ_M(M3+1))
1{1 \ 1 \ 5}2 has level ψ(I_4) = ψ(M4)
1{1 \ 1 \ 6}2 has level ψ(I_5) = ψ(M5)
1{1 \ 1 \ 1,2}2 has level ψ(I_ω) = ψ(Mω)
1{1 {1 \ 2} 2 \ 1 \ 1,2}2 has level ψ(I_ω+Ω)
1{1 {1 \ 1 \ 2} 2 \ 1 \ 1,2}2 has level ψ(I_ω+I)
1{1 {1 \ 1 \ 1,2} 2 \ 1 \ 1,2}2 has level ψ(I_ω*2)
1{1 {1 \ 1 \ 1,2} 1 {1 \ 1 \ 1,2} 2 \ 1 \ 1,2}2 has level ψ(I_ω^2)
1{1 {2 \ 1 \ 1,2} 2 \ 1 \ 1,2}2 has level ψ(I_ω^ω)
1{1 \ 2 \ 1,2}2 has level ψ(Ω(I_ω+1))
1{1 \ 1 \ 2,2}2 has level ψ(I_(ω+1)) = ψ(Mω+M)
1{1 \ 1 \ 3,2}2 has level ψ(I_(ω+2)) = ψ(Mω+M2)
1{1 \ 1 \ 1,3}2 has level ψ(I_(ω2)) = ψ(Mω2)
1{1 \ 1 \ 1,1,2}2 has level ψ(I_(ω^2)) = ψ(Mω^2)
1{1 \ 1 \ 1{2}2}2 has level ψ(I_(ω^ω)) = ψ(Mω^ω)
1{1 \ 1 \ 1 {1 \ 2} 2}2 has level ψ(I_(Ω)) = ψ(MΩ)
1{1 \ 1 \ 1 {1 \ 3} 2}2 has level ψ(I_(Ω_2)) = ψ(MΩ_2)
1{1 \ 1 \ 1 {1 \ 1,2} 2}2 has level ψ(I_(Ω_ω)) = ψ(MΩ_ω)
1{1 \ 1 \ 1 {1 \ 1 \ 2} 2}2 has level ψ(I_I) = ψ(Mψ_M(M))
1{1 \ 1 \ 1 {1 \ 1 \ 2} 2}2 has level ψ(I_Ω(I+1)) = ψ(Mψ_M(M+1))
1{1 \ 1 \ 1 {1 \ 1 \ 3} 2}2 has level ψ(I_(I_2)) = ψ(Mψ_M(M2))
1{1 \ 1 \ 1 {1 \ 1 \ 1,2} 2}2 has level ψ(I_(I_ω)) = ψ(Mψ_M(Mω))
1{1 \ 1 \ 1 {1 \ 1 \ 1 {1 \ 1 \ 2} 2} 2}2 has level ψ(I_(I_I)) = ψ(Mψ_M(Mψ_M(M)))
Then...
1{1 \ 1 \ 1 \ 2}2 has level ψ(I(2,0)) = ψ(M^2).
1{1 {1 \ 2} 2 \ 1 \ 1 \ 2}2 has level ψ(I(2,0)+Ω)
1{1 {1 \ 1 \ 2} 2 \ 1 \ 1 \ 2}2 has level ψ(I(2,0)+I)
1{1 {1 \ 1 \ 3} 2 \ 1 \ 1 \ 2}2 has level ψ(I(2,0)+I_2)
1{1 {1 \ 1 \ 1 {1 \ 1 \ 1 \ 2} 2} 2 \ 1 \ 1 \ 2}2 has level ψ(I(2,0)+ψ_{I(2,0)}(I(2,0)))
1{1 {1 \ 1 \ 1 \ 2} 2 \ 1 \ 1 \ 2}2 has level ψ(I(2,0)*2)
1{1 {1 \ 1 \ 1 \ 2} 1,2 \ 1 \ 1 \ 2}2 has level ψ(I(2,0)*ω)
1{1 {1 \ 1 \ 1 \ 2} 1 {1 \ 1 \ 1 \ 2} 2 \ 1 \ 1 \ 2}2 has level ψ(I(2,0)^2)
1{1 {2 \ 1 \ 1 \ 2} 2 \ 1 \ 1 \ 2}2 has level ψ(I(2,0)^ω)
1{1 {1 {1 \ 1 \ 1 \ 2} 2 \ 1 \ 1 \ 2} 2 \ 1 \ 1 \ 2}2 has level ψ(I(2,0)^I(2,0))
1{1 \ 2 \ 1 \ 2}2 has level ψ(Ω(I(2,0)+1)) = ψ(M^2+ψ_M(M^2+1))
1{1 \ 1,2 \ 1 \ 2}2 has level ψ(Ω(I(2,0)+ω)) = ψ(M^2+ψ_M(M^2+ω))
1{1 \ 1 {1 \ 1 \ 1 \ 2} 2 \ 1 \ 2}2 has level ψ(Ω(I(2,0)*2)) = ψ(M^2+ψ_M(M^2+ψ_M(M^2)))
1{1 \ 1 \ 2 \ 2}2 has level ψ(I(1,I(2,0)+1)) = ψ(M^2+M)
1{1 \ 2 \ 2 \ 2}2 has level ψ(Ω(I(1,I(2,0)+1)+1)) = ψ(M^2+M+ψ_M(M^2+M+1))
1{1 \ 1 \ 3 \ 2}2 has level ψ(I(1,I(2,0)+2)) = ψ(M^2+M2)
1{1 \ 1 \ 1,2 \ 2}2 has level ψ(I(1,I(2,0)+ω)) = ψ(M^2+Mω)
1{1 \ 1 \ 1 {1 \ 1 \ 1 \ 2} 2 \ 2}2 has level ψ(I(1,I(2,0)*2)) = ψ(M^2+Mψ(M^2))
1{1 \ 1 \ 1 \ 3}2 has level ψ(I(2,1)) = ψ(M^2*2)
1{1 \ 1 \ 1 \ 4}2 has level ψ(I(2,2)) = ψ(M^2*3)
1{1 \ 1 \ 1 \ 1,2}2 has level ψ(I(2,ω)) = ψ(M^2*ω)
1{1 \ 1 \ 1 \ 1 {1 \ 2} 2}2 has level ψ(I(2,Ω)) = ψ(M^2*Ω)
1{1 \ 1 \ 1 \ 1 {1 \ 1 \ 2} 2}2 has level ψ(I(2,I(1,0))) = ψ(M^2*ψ_M(M))
1{1 \ 1 \ 1 \ 1 {1 \ 1 \ 1 \ 2} 2}2 has level ψ(I(2,I(2,0))) = ψ(M^2*ψ_M(M^2))
1{1 \ 1 \ 1 \ 1 \ 2}2 has level ψ(I(3,0)) = ψ(M^3)
1{1 \ 1 \ 1 \ 1 \ 3}2 has level ψ(I(3,1)) = ψ(M^3*2)
1{1 \ 1 \ 1 \ 1 \ 1 \ 2}2 has level ψ(I(4,0)) = ψ(M^4)
1{1 \ 1 \ 1 \ 1 \ 1 \ 1 \ 2}2 has level ψ(I(5,0)) = ψ(M^5)
We can have dimensional \ arrays like (for example, nesting the previous level of \ separators, we can even have something like {1 \ 2}\, {2 \ 2}\, {1 \ 3}\, {1 \ 1 \ 2}\, {1 {2}\ 2}\, and so on):
1{1 {2}\ 2}2 has level ψ(I(ω,0)) = ψ(M^ω)
1{1 \ 2 {2}\ 2}2 has level ψ(Ω(I(ω,0)+1)) = ψ(M^ω+ψ_M(M^ω+1))
1{1 \ 1 \ 2 {2}\ 2}2 has level ψ(I(1,I(ω,0)+1)) = ψ(M^ω+M)
1{1 {2}\ 3}2 has level ψ(I(ω,1)) = ψ(M^ω*2)
1{1 {2}\ 1,2}2 has level ψ(I(ω,ω)) = ψ(M^ω*ω)
1{1 {2}\ 1 {1 {2}\ 2} 2}2 has level ψ(I(ω,ω)) = ψ(M^ω*ψ_M(M^ω))
1{1 {2}\ 1 \ 2}2 has level ψ(I(ω+1,0)) = ψ(M^(ω+1))
1{1 {2}\ 1 \ 1 \ 2}2 has level ψ(I(ω+2,0)) = ψ(M^(ω+2))
1{1 {2}\ 1 {2}\ 2}2 has level ψ(I(ω2,0)) = ψ(M^(ω2))
1{1 {3}\ 2}2 has level ψ(I(ω^2,0)) = ψ(M^(ω^2))
1{1 {4}\ 2}2 has level ψ(I(ω^3,0)) = ψ(M^(ω^3))
1{1 {1,2}\ 2}2 has level ψ(I(ω^ω,0)) = ψ(M^(ω^ω))
1{1 {1,1,2}\ 2}2 has level ψ(I(ω^ω^2,0)) = ψ(M^(ω^ω^2))
1{1 {1{2}2}\ 2}2 has level ψ(I(ω^ω^ω,0)) = ψ(M^(ω^ω^ω))
1{1 {1 {1 {1 \ 2} 2} 2}\ 2}2 has level ψ(I(ε0,0)) = ψ(M^(ψ(Ω)))
1{1 {1 {1 \ 2} 2}\ 2}2 has level ψ(I(Ω,0)) = ψ(M^Ω)
1{1 {1 {1 \ 3} 2}\ 2}2 has level ψ(I(Ω_2,0)) = ψ(M^(Ω_2))
1{1 {1 {1 \ 1 \ 2} 2}\ 2}2 has level ψ(I(I(1,0),0)) = ψ(M^(ψ_M(M)))
1{1 {1 {1 \ 1 \ 1 \ 2} 2}\ 2}2 has level ψ(I(I(2,0),0)) = ψ(M^(ψ_M(M^2)))
1{1 {1 {1 {2}\ 2} 2}\ 2}2 has level ψ(I(I(ω,0),0)) = ψ(M^(ψ_M(M^2)))
1{1 {1 {1 {1 {1 \ 2} 2}\ 2} 2}\ 2}2 has level ψ(I(I(Ω,0),0)) = ψ(M^(ψ_M(M^Ω)))
1{1 {1 {1 {1 {1 \ 1 \ 2} 2}\ 2} 2}\ 2}2 has level ψ(I(I(I(1,0),0),0)) = ψ(M^(ψ_M(M^(ψ_M(M)))))
Moving on...
1{1 {1 \ 2}\ 2}2 has level ψ(I(1,0,0)) = ψ(M^M)
1{1 \ 2 {1 \ 2}\ 2}2 has level ψ(Ω(I(1,0,0)+1)) = ψ(M^M+ψ_M(M^M+1))
1{1 \ 1 \ 2 {1 \ 2}\ 2}2 has level ψ(I(1,I(1,0,0)+1)) = ψ(M^M+M)
1{1 {2}\ 2 {1 \ 2}\ 2}2 has level ψ(I(ω,I(1,0,0)+1)) = ψ(M^M+M^ω)
1{1 {1 {1 \ 2} 2}\ 2 {1 \ 2}\ 2}2 has level ψ(I(Ω,I(1,0,0)+1)) = ψ(M^M+M^Ω)
1{1 {1 {1 {1 \ 2}\ 2} 2}\ 2 {1 \ 2}\ 2}2 has level ψ(I(I(1,0,0),1)) = ψ(M^M+M^(ψ_M(M^M)))
1{1 {1 \ 2}\ 3}2 has level ψ(I(1,0,1)) = ψ(M^M*2)
1{1 {1 \ 2}\ 1,2}2 has level ψ(I(1,0,ω)) = ψ(M^M*ω)
1{1 {1 \ 2}\ 1 {1 \ 2} 2}2 has level ψ(I(1,0,Ω)) = ψ(M^M*Ω)
1{1 {1 \ 2}\ 1 {1 \ 1 \ 2} 2}2 has level ψ(I(1,0,I)) = ψ(M^M*I)
1{1 {1 \ 2}\ 1 \ 2}2 has level ψ(I(1,1,0)) = ψ(M^(M+1))
1{1 {1 \ 2}\ 1 \ 1 \ 2}2 has level ψ(I(1,2,0)) = ψ(M^(M+2))
1{1 {1 \ 2}\ 1 {2}\ 2}2 has level ψ(I(1,ω,0)) = ψ(M^(M+ω))
1{1 {1 \ 2}\ 1 {1 {1 \ 1 \ 2} 2}\ 2}2 has level ψ(I(1,I,0)) = ψ(M^(M+I))
1{1 {1 \ 2}\ 1 {1 \ 2}\ 2}2 has level ψ(I(2,0,0)) = ψ(M^(M2))
1{1 {1 \ 2}\ 1 {1 \ 2}\ 1 {1 \ 2}\ 2}2 has level ψ(I(3,0,0)) = ψ(M^(M3))
1{1 {2 \ 2}\ 2}2 has level ψ(I(ω,0,0)) = ψ(M^(Mω))
1{1 {1 {1 \ 2} 2 \ 2}\ 2}2 has level ψ(I(Ω,0,0)) = ψ(M^(MΩ))
1{1 {1 {1 \ 1 \ 2} 2 \ 2}\ 2}2 has level ψ(I(I,0,0)) = ψ(M^(MI))
1{1 {1 \ 3}\ 2}2 has level ψ(I(1,0,0,0)) = ψ(M^M^2)
1{1 {1 \ 3}\ 1 {1 \ 3}\ 2}2 has level ψ(I(2,0,0,0)) = ψ(M^(M^2*2))
1{1 {1 \ 4}\ 2}2 has level ψ(I(1,0,0,0,0)) = ψ(M^M^3)
1{1 {1 \ 1,2}\ 2}2 has level ψ(M^M^ω)
1{1 {1 \ 1 {1 \ 2} 2}\ 2}2 has level ψ(M^M^Ω)
1{1 {1 \ 1 {1 \ 1 \ 2} 2}\ 2}2 has level ψ(M^M^I)
1{1 {1 \ 1 \ 2}\ 2}2 has level ψ(M^M^M)
1{1 {1 \ 1 \ 1 \ 2}\ 2}2 has level ψ(M^M^M^2)
1{1 {1 {1 \ 2}\ 2}\ 2}2 has level ψ(M^M^M^M)
1{1 {1 {1 \ 1 \ 2}\ 2}\ 2}2 has level ψ(M^M^M^M^M)
So the limit of the nested \ arrays is approximately ψ(ε(M+1)) = ψ(Ω(M+1)).
At this point, we need to define a separator that ranks higher than all the {x}\ separators. To do this, just use (\n) to diagonalize over the previous {x}\ separators. Of course, after \{{1}} comes \{{2}}, \{{1,2}}, \{{1{1 \ 2}2}}, \{{1 \ 2}}, and so on.
1{1 \{{1}} 2}2 has level ψ(Ω(M+1))
1{1 \ 2 \{{1}} 2}2 has level ψ(Ω(M+1)+M)
1{1 {1 \{{1}} 2}\ 2 \{{1}} 2}2 has level ψ(Ω(M+1)+ε(M+1))
1{1 \{{1}} 3}2 has level ψ(Ω(M+1)2)
1{1 \{{1}} 1,2}2 has level ψ(Ω(M+1)ω)
1{1 \{{1}} 1 \ 2}2 has level ψ(Ω(M+1)M)
1{1 \{{1}} 1 \{{1}} 2}2 has level ψ(Ω(M+1)^2)
1{1 {2}\{{1}} 2}2 has level ψ(Ω(M+1)^ω)
1{1 {1 \ 2}\{{1}} 2}2 has level ψ(Ω(M+1)^M)
1{1 {1 \{{1}} 2}\{{1}} 2}2 has level ψ(Ω(M+1)^Ω(M+1))
1{1 \{{2}} 2}2 has level ψ(Ω(M+2))
1{1 \{{3}} 2}2 has level ψ(Ω(M+3))
1{1 \{{4}} 2}2 has level ψ(Ω(M+4))
1{1 \{{1,2}} 2}2 has level ψ(Ω(M+ω))
1{1 \{{1 {1 \ 2} 2}} 2}2 has level ψ(Ω(M+Ω))
1{1 \{{1 {1 \ 1 \ 2} 2}} 2}2 has level ψ(Ω(M+I))
1{1 \{{1 {1 \{{1}} 2} 2}} 2}2 has level ψ(Ω(M+ψ_M(Ω(M+1))))
1{1 \{{1 \ 2}} 2}2 has level ψ(Ω(M2))
1{1 \{{2 \ 2}} 2}2 has level ψ(Ω(M2+1))
1{1 \{{1 \ 3}} 2}2 has level ψ(Ω(M3))
1{1 \{{1 \ 1,2}} 2}2 has level ψ(Ω(Mω))
1{1 \{{1 \ 1 \ 2}} 2}2 has level ψ(Ω(M^2))
1{1 \{{1 {2}\ 2}} 2}2 has level ψ(Ω(M^ω))
1{1 \{{1 {1 \ 2}\ 2}} 2}2 has level ψ(Ω(M^M))
1{1 \{{1 {1 \{{1}} 2}\ 2}} 2}2 has level ψ(Ω(ε(M+1)))
1{1 \{{1 \{{1}} 2}} 2}2 has level ψ(Ω(Ω(M+1)))
1{1 \{{1 \{{1 \ 2}} 2}} 2}2 has level ψ(Ω(Ω(M2)))
1{1 \{{1 \{{1 \{{1}} 2}} 2}} 2}2 has level ψ(Ω(Ω(Ω(M+1))))
The limit is ψ(I_(M+1)) = ψ(M_2)
And finally, moving on the multiple \ separators! Use the ^ symbol to reach even more. The array 1{1 \\ x}2 diagonalizes over the sequence 1{1 \{{x+1}} 2}2.
1{1 \\ 1 \\ 2}2 has level ψ(M_2) = ψ(I_(M+1))
1{1 \ 2 \\ 1 \\ 2}2 has level ψ(M_2+M) = ψ(I_(M+1)+M)
1{1 \{{1}} 2 \\ 1 \\ 2}2 = 1{1 {1 \\ 2} 2 \\ 1 \\ 2}2 has level ψ(M_2+Ω(M+1)) = ψ(I_(M+1)+Ω(M+1))
1{1 {1 \\ 1,2} 2 \\ 1 \\ 2}2 has level ψ(M_2+Ω(M+ω))
1{1 {1 \\ 1 {1 \\ 2} 2} 2 \\ 1 \\ 2}2 has level ψ(M_2+Ω(M2))
1{1 {1 \\ 1 {1 \\ 1 \\ 2} 2} 2 \\ 1 \\ 2}2 has level ψ(M_2+ψ_{I_(M+1)}(M_2))
1{1 {1 \\ 1 \\ 2} 2 \\ 1 \\ 2}2 has level ψ(M_2+ψ_{M_2}(M_2))
1{1 \\ 2 \\ 2}2 has level ψ(M_2+ψ_{M_2}(M_2+1))
1{1 \\ 1 \\ 3}2 has level ψ(M_2*2)
1{1 \\ 1 \\ 1,2}2 has level ψ(M_2*ω)
1{1 \\ 1 \\ 1 \ 2}2 has level ψ(M_2*M)
1{1 \\ 1 \\ 1 {1 \\ 2} 2}2 has level ψ(M_2*Ω(M+1))
1{1 \\ 1 \\ 1 \\ 2}2 has level ψ(M_2^2)
1{1 {2}\\ 2}2 has level ψ(M_2^ω)
1{1 {1 \ 2}\\ 2}2 has level ψ(M_2^M)
1{1 {1 \\ 2}\\ 2}2 has level ψ(M_2^M_2)
1{1 {1 \\ 1 \\ 2}\\ 2}2 has level ψ(M_2^M_2^M_2)
1{1 \\{{1}} 2}2 has level ψ(Ω(M_2+1))
1{1 \\{{1}} 3}2 has level ψ(Ω(M_2+1)*2)
1{1 \\{{1}} 1,2}2 has level ψ(Ω(M_2+1)*ω)
1{1 \\{{1}} 1 \\{{1}} 2}2 has level ψ(Ω(M_2+1)^2)
1{1 \\{{2}} 2}2 has level ψ(Ω(M_2+2))
1{1 \\{{1,2}} 2}2 has level ψ(Ω(M_2+ω))
1{1 \\{{1 \ 2}} 2}2 has level ψ(Ω(M_2+M))
1{1 \\{{1 \\ 2}} 2}2 has level ψ(Ω(M_2*2))
1{1 \\{{1 \\{{1}} 2}} 2}2 has level ψ(Ω(Ω(M_2+1)))
1{1 \\\ 1 \\\ 2}2 has level ψ(M_3) = ψ(I_(M_2+1))
1{1 \\\ 1 \\\ 3}2 has level ψ(M_3*2)
1{1 \\\ 1 \\\ 1 \\\ 2}2 has level ψ(M_3^2)
1{1 \\\{{1}} 2}2 has level ψ(Ω(M_3+1))
1{1 \\\\ 1 \\\\ 2}2 has level ψ(M_4)
1{1 \\\\\ 1 \\\\\ 2}2 has level ψ(M_5)
Now we need a way to generalize the last section. Here, I use a caret (^) symbol after the \ to denote the number of slashes. So \^1 = \, \^2 = \\, \^3 = \\\, and so on. Of course, you can even have \^(a){{b}} but these aren't necessary. To finish up, we can nest the slashes after the ^.
1{1 \^(1,2) 2}2 has level ψ(M_ω)
1{1 \^(1,2) 1 \^(1,2) 2}2 has level ψ(M_ω^2)
1{1 \^(2,2) 2}2 has level ψ(Ω(M_ω+1))
1{1 \^(2,2) 2 \^(2,2) 2}2 has level ψ(M_(ω+1))
1{1 \^(1,3) 2}2 has level ψ(M_(ω2))
1{1 \^(1,1,2) 2}2 has level ψ(M_(ω^2))
1{1 \^(1{2}2) 2}2 has level ψ(M_(ω^ω))
1{1 \^(1 {1 {1 \ 2} 2} 2) 2}2 has level ψ(M_(ε0))
1{1 \^(1 {1 \ 2} 2) 2}2 has level ψ(M_Ω)
1{1 \^(1 {1 \ 1 \ 2} 2) 2}2 has level ψ(M_I)
1{1 \^(1 {1 \ 1 \ 2} 2) 2}2 has level ψ(M_I)
1{1 \^(1 \ 2) 2}2 has level ψ(M_M)
1{1 \^(1 \\ 2) 2}2 has level ψ(M_M_2)
1{1 \^(1 \^(1,2) 2) 2}2 has level ψ(M_M_ω)
1{1 \^(1 \^(1 {1 \ 2} 2) 2) 2}2 has level ψ(M_M_Ω)
1{1 \^(1 \^(1 {1 \ 1 \ 2} 2) 2) 2}2 has level ψ(M_M_I)
1{1 \^(1 \^(1 \ 2) 2) 2}2 has level ψ(M_M_M)
1{1 \^(1 \^(1 \\ 2) 2) 2}2 has level ψ(M_M_M_2)
1{1 \^(1 \^(1 \^(1,2) 2) 2) 2}2 has level ψ(M_M_M_ω)
1{1 \^(1 \^(1 \^(1 \ 2) 2) 2) 2}2 has level ψ(M_M_M_M)
1{1 \^(1 \^(1 \^(1 \^(1,2) 2) 2) 2) 2}2 has level ψ(M_M_M_M_M)
So the limit of the Extended Expanding Array Notation is ψ(M(2,0)), which is the Mahlo fixed point (ψ(M_M_M_...)). It is also equal to ψ(N) = ψ(M(2;0)).
---<| THE END |>---
Update 1st January 2023, following the introduction of New Year's Numbers 2023:
I define the special function that diagonalizes over the limit of the notation, called M function, to be:
M(a, 0) = a[1, 2]a = a[a]a
M(a, 1) = a[1 {1 \ 2} 2]a = a[1 {1 / 2} 2]a
M(a, 2) = a[1 {1 \^(1 \ 2) 2} 2]a
M(a, 3) = a[1 {1 \^(1 \^(1 \ 2) 2) 2} 2]a
M(a, 4) = a[1 {1 \^(1 \^(1 \^(1 \ 2) 2) 2) 2} 2]a
M(a, b) = a[1 {1 \^(1 \^(1 \^(... 1 \^(1 \^(1 \^(1 \ 2) 2) 2) 2 ...) 2) 2) 2} 2]a w/ b backslashes (\), for b >= 1