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A089850 Involution of natural numbers induced by Catalan automorphism *A089850 acting on the binary trees/parenthesizations encoded by A014486/A063171.

%I A089850 #11 Mar 31 2012 14:02:27
%S A089850 0,1,2,3,5,4,6,7,8,12,13,11,9,10,15,14,16,17,18,19,20,21,22,31,32,34,
%T A089850 35,36,30,33,28,23,24,29,25,26,27,40,41,39,37,38,43,42,44,45,46,47,48,
%U A089850 49,50,52,51,53,54,55,56,57,58,59,60,61,62,63,64,87,88,90,91,92,96,97,99
%N A089850 Involution of natural numbers induced by Catalan automorphism *A089850 acting on the binary trees/parenthesizations encoded by A014486/A063171.
%C A089850 This automorphism effects the following transformation on the unlabeled rooted plane binary trees (letters A, B, C refer to arbitrary subtrees located on those nodes and () stands for an implied terminal node.)
%C A089850 ...B...C...........C...B
%C A089850 ....\./.............\./
%C A089850 .A...x....-->....A...x.................A..().........A...()..
%C A089850 ..\./.............\./...................\./....-->....\./...
%C A089850 ...x...............x.....................x.............x....
%C A089850 (a . (b . c)) -> (a . (c . b)) ______ (a . ()) ---> (a . ())
%C A089850 In terms of S-expressions, this automorphism swaps cadr and cddr of an S-exp if its length > 1.
%C A089850 Look at the example in A069770 to see how this will produce the given sequence of integers.
%H A089850 A. Karttunen, <a href="http://oeis.org/wiki/Catalan_Automorphisms">Catalan Automorphisms</a>
%H A089850 A. Karttunen, <a href="/A089408/a089408.c.txt">C-program for computing this sequence</a>
%H A089850 <a href="/index/Per#IntegerPermutationCatAuto">Index entries for signature-permutations induced by Catalan automorphisms</a>
%o A089850 (Scheme functions implementing this automorphism on list-structures/S-expressions, constructive (*A089850) and destructive (*A089850!) version:)
%o A089850 (define (*A089850 s) (if (and (pair? s) (pair? (cdr s))) (cons (car s) (cons (cddr s) (cadr s))) s))
%o A089850 (define (*A089850! s) (cond ((not (pair? s)) s) ((not (pair? (cdr s))) s) (else (*A069770! (cdr s)) s)))
%Y A089850 a(n) = A069770(A089859(n)) = A089863(A069770(n)) = A057163(A089854(A057163(n))). Row 3 of A089840. Row 3771 of A122203 and row 3677 of A122204.
%Y A089850 Number of cycles: A073191. Number of fixed points: A073190. Max. cycle size & LCM of all cycle sizes: A046698 (in each range limited by A014137 and A014138).
%K A089850 nonn
%O A089850 0,3
%A A089850 _Antti Karttunen_, Nov 29 2003
%E A089850 The new mail-address, a graphical explanation and constructive implementation of Scheme-function (*A089850) added by _Antti Karttunen_, Jun 04 2011