%I
%S 0,1,2,3,4,5,8,7,6,9,10,11,12,13,21,22,20,17,18,19,16,14,15,23,24,25,
%T 26,27,28,29,30,31,32,33,34,35,36,58,59,62,63,64,57,61,54,45,46,55,48,
%U 49,50,56,60,53,44,47,51,42,37,38,52,43,39,40,41,65,66,67,68,69,70,71,72
%N Involution of natural numbers induced by Catalan Automorphism *A089856 acting on the binary trees/parenthesizations encoded by A014486/A063171.
%C 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 .A...B...........C...B
%C ..\./.............\./
%C ...x...C....>....x...A...............()..A.........()..A..
%C ....\./.............\./.................\./....>....\./...
%C .....x...............x...................x.............x....
%C ((a . b) . c) > ((c . b) . a) _____ (() . a) > (() . a)
%C In terms of Sexpressions, this automorphism swaps caar and cdr of an Sexp if possible, i.e., if carside is not ().
%C See the Karttunen OEISWiki link for a detailed explanation of how to obtain a given integer sequence from this definition.
%H A. Karttunen, <a href="http://oeis.org/wiki/Catalan_Automorphisms">Catalan Automorphisms</a>
%H A. Karttunen, <a href="/A089408/a089408.c.txt">Cprogram for computing this sequence</a>
%H <a href="/index/Per#IntegerPermutationCatAuto">Index entries for signaturepermutations induced by Catalan automorphisms</a>
%o (Scheme function implementing this automorphism on liststructures/Sexpressions, both constructive (*A089856) and destructive (*A089856!) versions:)
%o (define (*A089856 s) (if (and (pair? s) (pair? (car s))) (cons (cons (cdr s) (cdar s)) (caar s)) s))
%o (define (*A089856! s) (cond ((not (pair? s)) s) ((not (pair? (car s))) s) (else (let ((org_caar (caar s))) (setcar! (car s) (cdr s)) (setcdr! s org_caar) s))))
%Y Row 10 of A089840. a(n) = A073269(A069770(n)) = A069770(A073270(n)) = A057163(A089852(A057163(n))).
%Y 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 nonn
%O 0,3
%A _Antti Karttunen_, Nov 29 2003
%E Further comments and constructive implementation of Schemefunction (*A089856) added by _Antti Karttunen_, Jun 04 2011
