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 A069768 Signature-permutation of Catalan bijection "Knack". 32

%I

%S 0,1,3,2,8,7,6,4,5,22,21,20,17,18,19,16,14,9,10,15,11,12,13,64,63,62,

%T 58,59,61,57,54,45,46,55,48,49,50,60,56,53,44,47,51,42,37,23,24,38,25,

%U 26,27,52,43,39,28,29,40,30,31,32,41,33,34,35,36,196,195,194,189,190

%N Signature-permutation of Catalan bijection "Knack".

%C This automorphism of binary trees first swaps the left and right subtree of the root and then proceeds recursively to the (new) left subtree, to do the same operation there. This is one of those Catalan bijections which extend to a unique automorphism of the infinite binary tree, which in this case is A153142. See further comments there and in A153141.

%C This bijection, Knack, is a ENIPS-transformation of the simple swap: ENIPS(*A069770) (i.e., row 1 of A122204). Furthermore, Knack and Knick (the inverse, A069767) have a special property, that FORK and KROF transforms (explained in A122201 and A122202) transform them to their own inverses, i.e., to each other: FORK(Knick) = KROF(Knick) = Knack and FORK(Knack) = KROF(Knack) = Knick, thus this occurs also as row 1 in A122288 and naturally, the double-fork fixes both, e.g., FORK(FORK(Knack)) = Knack.

%C Note: the name in Finnish is "Naks".

%D A. Karttunen, paper in preparation.

%H A. Karttunen, <a href="/A069768/b069768.txt">Table of n, a(n) for n = 0..2055</a>

%H <a href="/index/Per#IntegerPermutationCatAuto">Index entries for signature-permutations induced by Catalan bijections</a>

%o (Scheme implementations of this automorphism. These act on S-expressions, i.e. list-structures:)

%o (CONSTRUCTIVE VERSION:) (define (*A069768 s) (cond ((not (pair? s)) s) (else (cons (*A069768 (cdr s)) (car s)))))

%o (DESTRUCTIVE VERSION:) (define (*A069768! s) (cond ((pair? s) (*A069768! (cdr s)) (*A069770! s))) s)

%Y Inverse permutation: "Knick", A069767. "n-th powers" (i.e. n-fold applications), from n=2 to 6: A073291, A073293, A073295, A073297, A073299.

%Y In range [A014137(n-1)..A014138(n-1)] of this permutation, the number of cycles is A073431, number of fixed points: A036987 (Fixed points themselves: A084108), Max. cycle size & LCM of all cycle sizes: A011782. See also: A074080.

%Y A127302(a(n)) = A127302(n) for all n. a(n) = A057162(A057508(n)) = A069769(A057162(n))

%Y Row 1 of A122204 and A122288, row 21 of A122285 and A130402, row 8 of A073200.