This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
This involution (self-inverse permutation) of natural numbers is induced when we reflect the rooted plane binary trees encoded by A014486. E.g., we have A014486(5) = 44 (101100 in binary), A014486(7) = 52 (110100 in binary) and these encode the following rooted plane binary trees, which are reflections of each other:
0 0 0 0
\ / \ /
1 0 0 1
\ / \ /
0 1 1 0
\ / \ /
1 1
thus a(5)=7 and a(7)=5.
a(n) = A080300(ReflectBinTree(A014486(n))) ReflectBinTree := n -> ReflectBinTree2(n)/2; ReflectBinTree2 := n -> (`if`((0 = n),n,ReflectBinTreeAux(A030101(n)))); ReflectBinTreeAux := proc(n) local a,b; a := ReflectBinTree2(BinTreeLeftBranch(n)); b := ReflectBinTree2(BinTreeRightBranch(n)); RETURN((2^(A070939(b)+A070939(a))) + (b * (2^(A070939(a)))) + a); end; NextSubBinTree := proc(nn) local n,z,c; n := nn; c := 0; z := 0; while(c < 1) do z := 2*z + (n mod 2); c := c + (-1)^n; n := floor(n/2); od; RETURN(z); end; BinTreeLeftBranch := n -> NextSubBinTree(floor(n/2)); BinTreeRightBranch := n -> NextSubBinTree(floor(n/(2^(1+A070939(BinTreeLeftBranch(n))))));
A014486Q[0] = True; A014486Q[n_] := Catch[Fold[If[# < 0, Throw[False], If[#2 == 0, # - 1, # + 1]] &, 0, IntegerDigits[n, 2]] == 0]; tree[n_] := Block[{func, num = Append[IntegerDigits[n, 2], 0]}, func := If[num[[1]] == 0, num = Drop[num, 1]; 0, num = Drop[num, 1]; 1[func, func]]; func]; A057163L[n_] := Function[x, FirstPosition[x, FromDigits[Most@Cases[tree[#] /. 1 -> Reverse@*1, 0 | 1, All, Heads -> True], 2]][[1]] - 1 & /@ x][Select[Range[0, 2^n], A014486Q]]; A057163L[11] (* JungHwan Min, Dec 11 2016 *)
To obtain the signature permutation, we apply these transformations to the binary trees as encoded and ordered by A014486 and for each n, a(n) will be the position of the tree to which the n-th tree is transformed to, as follows:
.
one tree of one internal
empty tree (non-leaf) node
x \/
n= 0 1
a(n)= 0 1 (both are always fixed)
.
the next 7 trees, with 2-3 internal nodes, in range [A014137(1), A014137(2+1)-1] = [2,8] are:
.
\/ \/ \/ \/
\/ \/ \/ \/ \/ \/ \/ \/
\/ \/ \/ \/ \_/ \/ \/
n= 2 3 4 5 6 7 8
.
and the new shapes after swapping their left and right hand subtrees are:
.
\/ \/ \/ \/
\/ \/ \/ \/ \/ \/ \/ \/
\/ \/ \/ \/ \_/ \/ \/
a(n)= 3 2 7 8 6 4 5
thus we obtain the first nine terms of this sequence: 0, 1, 3, 2, 7, 8, 6, 4, 5.
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