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A089855 Permutation of natural numbers induced by Catalan Automorphism *A089855 acting on the binary trees/parenthesizations encoded by A014486/A063171.

%I A089855 #22 May 01 2018 02:56:13
%S A089855 0,1,2,3,4,5,7,8,6,9,10,11,12,13,17,18,20,21,22,16,19,14,15,23,24,25,
%T A089855 26,27,28,29,30,31,32,33,34,35,36,45,46,48,49,50,54,55,57,58,59,61,62,
%U A089855 63,64,44,47,53,56,60,42,51,37,38,43,52,39,40,41,65,66,67,68,69,70,71,72
%N A089855 Permutation of natural numbers induced by Catalan Automorphism *A089855 acting on the binary trees/parenthesizations encoded by A014486/A063171.
%C A089855 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 A089855 .A...B...........B...C
%C A089855 ..\./.............\./
%C A089855 ...x...C....-->....x...A...............()..A.........()..A..
%C A089855 ....\./.............\./.................\./....-->....\./...
%C A089855 .....x...............x...................x.............x....
%C A089855 ((a . b) . c) -> ((b . c) . a) _____ (() . a) ---> (() . a)
%C A089855 In terms of S-expressions, this automorphism rotates caar, cdar and cdr of an S-exp if possible, i.e., if car-side is not ().
%C A089855 See the Karttunen OEIS-Wiki link for a detailed explanation of how to obtain a given integer sequence from this definition.
%H A089855 A. Karttunen, <a href="http://oeis.org/wiki/Catalan_Automorphisms">Catalan Automorphisms</a>
%H A089855 A. Karttunen, <a href="/A089408/a089408.c.txt">C-program for computing this sequence</a>
%H A089855 <a href="/index/Per#IntegerPermutationCatAuto">Index entries for signature-permutations induced by Catalan automorphisms</a>
%o A089855 (Scheme functions implementing this automorphism on list-structures/S-expressions, both constructive (*A089855) and destructive (*A089855!) versions:)
%o A089855 (define (*A089855 s) (if (and (pair? s) (pair? (car s))) (cons (cons (cdar s) (cdr s)) (caar s)) s))
%o A089855 (define (*A089855! s) (cond ((not (pair? s)) s) ((not (pair? (car s))) s) (else (let ((org_cdar (cdar s))) (set-cdr! (car s) (cdr s)) (set-cdr! s (caar s)) (set-car! (car s) org_cdar) s))))
%Y A089855 Inverse of A089857. a(n) = A089860(A069770(n)) = A069770(A074680(n)) = A057163(A089853(A057163(n))). Row 9 of A089840.
%Y A089855 Number of cycles: A089847. Number of fixed-points: A089848 (in each range limited by A014137 and A014138).
%K A089855 nonn
%O A089855 0,3
%A A089855 _Antti Karttunen_, Nov 29 2003
%E A089855 Further comments and constructive implementation of Scheme-function (*A089855) added by _Antti Karttunen_, Jun 04 2011