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A069767 Signature-permutation of Catalan bijection "Knick".

%I A069767 #11 Apr 01 2017 19:49:22
%S A069767 0,1,3,2,7,8,6,5,4,17,18,20,21,22,16,19,15,12,13,14,11,10,9,45,46,48,
%T A069767 49,50,54,55,57,58,59,61,62,63,64,44,47,53,56,60,43,52,40,31,32,41,34,
%U A069767 35,36,42,51,39,30,33,38,29,26,27,37,28,25,24,23,129,130,132,133,134
%N A069767 Signature-permutation of Catalan bijection "Knick".
%C A069767 This automorphism of binary trees first swaps the left and right subtree of the root and then proceeds recursively to the (new) right 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 A153141. See further comments there.
%C A069767 This bijection, Knick, is a SPINE-transformation of the simple swap: SPINE(*A069770) (i.e., row 1 of A122203). Furthermore, Knick and Knack (the inverse, *A069768) 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 a row 1 in A122287 and naturally, the double-fork fixes both, e.g., FORK(FORK(Knick)) = Knick. There are also other peculiar properties.
%C A069767 Note: the name in Finnish is "Niks".
%D A069767 A. Karttunen, paper in preparation.
%H A069767 A. Karttunen, <a href="/A069767/b069767.txt">Table of n, a(n) for n = 0..2055</a>
%H A069767 <a href="/index/Per#IntegerPermutationCatAuto">Index entries for signature-permutations induced by Catalan bijections</a>
%o A069767 (Scheme implementations of this automorphism. These act on S-expressions, i.e. list-structures:)
%o A069767 (CONSTRUCTIVE VERSION:) (define (*A069767 s) (cond ((not (pair? s)) s) (else (cons (cdr s) (*A069767 (car s))))))
%o A069767 (DESTRUCTIVE VERSION:) (define (*A069767! s) (cond ((pair? s) (*A069770! s) (*A069767! (cdr s)))) s)
%Y A069767 Inverse permutation: "Knack", A069768. "n-th powers" (i.e. n-fold applications), from n=2 to 6: A073290, A073292, A073294, A073296, A073298.
%Y A069767 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 A069767 A127302(a(n)) = A127302(n) for all n. a(n) = A057508(A057161(n)) = A057161(A069769(n)).
%Y A069767 Row 1 of A122203 and A122287, row 15 of A122286 and A130403, row 6 of A073200.
%Y A069767 See also bijections A073286, A082345, A082348, A082349, A130341.
%K A069767 nonn
%O A069767 0,3
%A A069767 _Antti Karttunen_, Apr 16 2002; entry revised Dec 20 2008