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On some comments on blog posts members of the community said that the w/ operator of HAN is difficult to understand, and I'm going to change that in this blog post. After that, I will explain some similarities of HAN with BEAF.

2 array chain

a![1]w/[1] expands to a![1,1...1,1,2] with a 1's

If the first array isn't [1], you must solve it using the normal way.

3![2]w/[1] expands to ((3![1]w/[1])![1]w/[1])![1]w/[1]

3![1,2]w/[1] expands to 3![[1,1]w/[1],1]w/[1] which expands to 3![[1]w/[1]]w/[1]

3![1,1,2]w/[1] expands to 3![1,[1,1,1]w/[1],1]w/[1] which expands to 3![1,[1]w/[1]]w/[1]

3![1,2,2]w/[1] expands to 3![[1,1,2]w/[1],1,2]w/[1]

If the first array is [1], and the second is not [1]:

a![1]w/[2] expands to a![1,2]w/[1]

a![1]w/[3] expands to a![1,1,2]w/[1]

a![1]w/[1,2] expands to a![1]w/[[1]w/[1]], which expands to a![1]w/[[1,1...1,1,2]]

If the first entry of the second array is bigger than 1, put it in the first array. Otherwise, solve the array.

Larger chain

a![1]w/[1]w/[1] expands to a![1]w/[1,1...1,1,2] with a a's

If the first array isn't [1], solve that linear array.

a![1,2]w/[1]w/[1] expands to a![[1]w/[1]w/[1]]w/[1]w/[1]

If the first array is [1], but the second isn't, solve that array.

a![1]w/[2]w/[1] expands to a![1,1...1,1,2]w/[1]w/[1]

a![1]w/[1,2]w/[1] expands to a![1]w/[[1]w/[1]w/[1]]w/[1]

If the first and the second array are [1], but the third isn't, solve that array.

a![1]w/[1]w/[1,2] expands to a![1]w/[1,1...1,1,2]w/[1]

If the first, the second array and the third array are [1], but the fourth isn't, solve that array.

a![1]w/[1]w/[1]w/[1] expands to a![1]w/[1]w/[1,1...1,1,2]

etc.

2 row arrays

a![1(1)2] expands to a![1]w/[1]w/[1]...[1]w/[1]w/[1]

a![1,2(1)2] expands to a![[1(1)2](1)2]

a![1(1)2]w/[1] (the w/ operator works in the first row) expands to a![1,1...1,1,2(1)2]

a![1(1)2]w/[1(1)2] (the w/ operator works in the first row) expands to a![1(1)2]w/[1]w/[1]...[1]w/[1]w/[1]

a![1(1)2]w/[2(1)2] (the w/ operator works in the first row) expands to a![1,2(1)2]w/[1(1)2]

a![1(1)2]w/[1(1)2]w/[1] (the w/ operator works in the first row) expands to a![1(1)2]w/[1,1...1,1,2(1)2]

a![1(1)3] expands to a![1(1)2]w/[1(1)2]w/[1(1)2]...[1(1)2]w/[1(1)2]w/[1(1)2]

a![1(1)k+1] expands to a![1(1)k]w/[1(1)k]w/[1(1)k]...[1(1)k]w/[1(1)k]w/[1(1)k]

a![1(1)1,2] expands to a![1(1)[1]]

a![1(1)1,1,2] expands to a![1(1)1,[1]]

a![1(1)1,2,2] expands to a![1(1)[1(1)1,1,2],1,2]

a![1(1)1]w/[1] (the w/ operator works in the second row) expands to a![1(1)1,1...1,1,2]

a![1(1)1]w/[1(1)2] (the w/ operator works in the second row) expands to a![1(1)1]w/[1(1)2]

a![1(1)1]w/[1(1)2] (the w/ operator works in the second row) expands to a![1(1)1]w/[1(1)2]

a![1(1)1(1)2] expands to a![1(1)1]w/[1(1)1]w/[1(1)1]...[1(1)1]w/[1(1)1]w/[1(1)1]

Planar arrays

a![1(1)1(1)2] expands to a![1(1)1]w/[1(1)1]w/[1(1)1]...[1(1)1]w/[1(1)1]w/[1(1)1]

a![1(1)1(1)3] expands to a![1(1)1(1)2]w/[1(1)1(1)2]w/[1(1)1(1)2]...[1(1)1(1)2]w/[1(1)1(1)2]w/[1(1)1(1)2]

a![1(1)1(1)1(1)2] expands to a![1(1)1(1)1]w/[1(1)1(1)1]w/[1(1)1(1)1]...[1(1)1(1)1]w/[1(1)1(1)1]w/[1(1)1(1)1]

a![1]w/(1)[1] expands to a![1(1)1...1(1)1(1)2] a 1's

a![1(2)2] expands to a![1]w(1)/[1]w(1)/[1]w(1)/[1]...[1]w(1)/[1]w(1)/[1]w(1)/[1]

a![1(2)3] expands to a![1(2)2]w(1)/[1(2)2]w(1)/[1(2)2]w(1)/[1(2)2]...[1(2)2]w(1)/[1(2)2]w(1)/[1(2)2]w(1)/[1(2)2]

a![1(2)k+1] expands to a![1(2)k]w(1)/[1(2)k]w(1)/[1(2)k]...[1(2)k]w(1)/[1(2)k]w(1)/[1(2)k]

etc.

Some facts

  • In the bracket types, the strength of the ground system doesn't matter much, even if the initial comparisons are very different:
HAN 1 FGH HAN 2 FGH
[1(1)[21]] \(\vartheta(\Omega+1)\) [1([21])2] \(\vartheta(\Omega^\Omega)\)
[1(1)[21,1,2]] \(\vartheta(\Omega+2)\) [1([21,1,2])2] \(\vartheta(\Omega^{\Omega^2})\)
[1(1)[21,1,1,2]] \(\vartheta(\Omega2)\) [1([21,1,1,2])2] \(\vartheta(\Omega^{\Omega^\Omega})\)
[1(1)[21,1,2,2]] \(\vartheta(\Omega^\Omega)\) [1([21,1,2,2])2] \(\vartheta(\Omega^{\Omega^{\Omega^\Omega}})\)
[1(1)[21,1,1,1,2]] \(\vartheta(\varepsilon_{\Omega+1})\) [1([21,1,1,1,2])2] \(\vartheta(\varepsilon_{\Omega+1})\)
  • The w/ operator is like the third entry in BEAF:
    • [1]w/[1] ~ {X,X,X}&n,
    • [1]w/[1,2] ~ {X,X,X2}&n
    • [1]w/[1]w/[1] ~ {X,X,{X,X,X}}&n
  • The w(1)/ operator is like the fourth entry in BEAF:
    • [1]w(1)/[1] ~ {X,X,X,X}&n
    • [1]w(1)/[1,2] ~ {X,X,X,X2}&n
    • [1]w(1)/[1]w(1)/[1] ~ {X,X,X,{X,X,X,X}}&n
  • The w(k)/ operator is like the (k+3)th entry in BEAF
  • The dimensional arrays in BEAF nests the level in seperators, and therefore {X,X,2(1)2}&n ~  [1([21])2]
  • If you continue you'll see the bracket types work the same as Xk
  • In HAN, the brackets with a quill work like my &n brackets

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