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A089859 Permutation of natural numbers induced by Catalan Automorphism *A089859 acting on the binary trees/parenthesizations encoded by A014486/A063171.

%I A089859 #20 May 01 2018 02:57:24
%S A089859 0,1,3,2,8,7,6,4,5,21,22,20,17,18,19,16,14,9,10,15,11,12,13,58,59,62,
%T A089859 63,64,57,61,54,45,46,55,48,49,50,56,60,53,44,47,51,42,37,23,24,38,25,
%U A089859 26,27,52,43,39,28,29,40,30,31,32,41,33,34,35,36,170,171,174,175,176,184
%N A089859 Permutation of natural numbers induced by Catalan Automorphism *A089859 acting on the binary trees/parenthesizations encoded by A014486/A063171.
%C A089859 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 A089859 .....B...C.......C...B
%C A089859 ......\./.........\./
%C A089859 ...A...x...-->... .x...A...............A..().........()..A..
%C A089859 ....\./.............\./.................\./....-->....\./...
%C A089859 .....x...............x...................x.............x....
%C A089859 (a . (b . c)) --> ((c . b) . a) _____ (a . ()) --> (() . a)
%C A089859 See the Karttunen OEIS-Wiki link for a detailed explanation of how to obtain a given integer sequence from this definition.
%H A089859 A. Karttunen, <a href="http://oeis.org/wiki/Catalan_Automorphisms">Catalan Automorphisms</a>
%H A089859 A. Karttunen, <a href="/A089408/a089408.c.txt">C-program for computing this sequence</a>
%H A089859 <a href="/index/Per#IntegerPermutationCatAuto">Index entries for signature-permutations induced by Catalan automorphisms</a>
%o A089859 (Scheme functions implementing this automorphism on list-structures/S-expressions, both constructive (*A089859) and destructive (*A089859!) versions:)
%o A089859 (define (*A089859 s) (cond ((not (pair? s)) s) ((not (pair? (cdr s))) (cons (cdr s) (car s))) (else (cons (cons (cddr s) (cadr s)) (car s)))))
%o A089859 (define (*A089859! s) (cond ((pair? s) (cond ((pair? (cdr s)) (*A069770! (cdr s)) (*A069770! s)) (else (*A069770! s))))) s)
%Y A089859 Row 15 of A089840. Inverse of A089863. a(n) = A089854(A069770(n)) = A069770(A089850(n)). A089864 is the "square" of this permutation.
%Y A089859 Number of cycles: A089407. Max. cycle size & LCM of all cycle sizes: A040002 (in each range limited by A014137 and A014138).
%K A089859 nonn
%O A089859 0,3
%A A089859 _Antti Karttunen_, Nov 29 2003
%E A089859 A graphical description and constructive implementation of Scheme-function (*A089859) added by _Antti Karttunen_, Jun 04 2011