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A089863 Permutation of natural numbers induced by Catalan Automorphism *A089863 acting on the binary trees/parenthesizations encoded by A014486/A063171.

%I A089863 #21 May 01 2018 02:57:02
%S A089863 0,1,3,2,7,8,6,5,4,17,18,20,21,22,16,19,15,12,13,14,11,9,10,45,46,48,
%T A089863 49,50,54,55,57,58,59,61,62,63,64,44,47,53,56,60,43,52,40,31,32,41,34,
%U A089863 35,36,42,51,39,30,33,37,28,23,24,38,29,25,26,27,129,130,132,133,134
%N A089863 Permutation of natural numbers induced by Catalan Automorphism *A089863 acting on the binary trees/parenthesizations encoded by A014486/A063171.
%C A089863 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 A089863 .A...B...............B...A
%C A089863 ..\./.................\./
%C A089863 ...x...C...-->.....C...x...............()..A.........A..()..
%C A089863 ....\./.............\./.................\./....-->....\./...
%C A089863 .....x...............x...................x.............x....
%C A089863 ((a . b) . c) --> (c . (b . a)) ____ (() . a) ----> (a . ())
%C A089863 See the Karttunen OEIS-Wiki link for a detailed explanation of how to obtain a given integer sequence from this definition.
%H A089863 A. Karttunen, <a href="http://oeis.org/wiki/Catalan_Automorphisms">Catalan Automorphisms</a>
%H A089863 A. Karttunen, <a href="/A089408/a089408.c.txt">C-program for computing this sequence</a>
%H A089863 <a href="/index/Per#IntegerPermutationCatAuto">Index entries for signature-permutations induced by Catalan automorphisms</a>
%o A089863 (Scheme functions implementing this automorphism on list-structures:/S-expressions, both constructive (*A089863) and destructive (*A089863!) versions:)
%o A089863 (define (*A089863 s) (cond ((not (pair? s)) s) ((not (pair? (car s))) (cons (cdr s) (car s))) (else (cons (cdr s) (cons (cdar s) (caar s))))))
%o A089863 (define (*A089863! s) (cond ((pair? s) (cond ((pair? (car s)) (*A069770! (car s)) (*A069770! s)) (else (*A069770! s))))) s)
%Y A089863 Row 21 of A089840. Inverse of A089859. a(n) = A089850(A069770(n)) = A069770(A089854(n)). A089864 is the "square" of this permutation.
%Y A089863 Number of cycles: A089407. Max. cycle size & LCM of all cycle sizes: A040002 (in each range limited by A014137 and A014138).
%K A089863 nonn
%O A089863 0,3
%A A089863 _Antti Karttunen_, Nov 29 2003
%E A089863 A graphical description and constructive version of Scheme-implementation added by _Antti Karttunen_, Jun 04 2011