cp's OEIS Frontend

A089852 Involution of natural numbers induced by Catalan automorphism *A089852 acting on the binary trees/parenthesizations encoded by A014486/A063171.

%I A089852 #23 May 01 2018 02:56:06
%S A089852 0,1,2,3,6,5,4,7,8,16,19,15,12,13,14,11,9,17,18,10,20,21,22,44,47,53,
%T A089852 56,60,43,52,40,31,32,41,34,35,36,42,51,39,30,33,37,28,23,45,46,24,48,
%U A089852 49,50,38,29,25,54,55,26,57,58,59,27,61,62,63,64,128,131,137,140,144
%N A089852 Involution of natural numbers induced by Catalan automorphism *A089852 acting on the binary trees/parenthesizations encoded by A014486/A063171.
%C A089852 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 A089852 ...B...C...........B...A
%C A089852 ....\./.............\./
%C A089852 .A...x....-->....C...x.................A..().........A...()..
%C A089852 ..\./.............\./...................\./....-->....\./...
%C A089852 ...x...............x.....................x.............x....
%C A089852 (a . (b . c)) -> (c . (b . a)) ______ (a . ()) ---> (a . ())
%C A089852 In terms of S-expressions, this automorphism swaps car and cddr of an S-exp if its length > 1, if possible, otherwise keeps it intact.
%C A089852 See the Karttunen OEIS-Wiki link for a detailed explanation of how to obtain a given integer sequence from this definition.
%H A089852 A. Karttunen, <a href="http://oeis.org/wiki/Catalan_Automorphisms">Catalan Automorphisms</a>
%H A089852 A. Karttunen, <a href="/A089408/a089408.c.txt">C-program for computing this sequence</a>
%H A089852 <a href="/index/Per#IntegerPermutationCatAuto">Index entries for signature-permutations induced by Catalan automorphisms</a>
%o A089852 (Scheme functions implementing this automorphism on list-structures/S-expressions, both constructive (*A089852) and destructive (*A089852!) versions:)
%o A089852 (define (*A089852 s) (if (and (pair? s) (pair? (cdr s))) (cons (cddr s) (cons (cadr s) (car s))) s))
%o A089852 (define (*A089852! s) (cond ((not (pair? s)) s) ((not (pair? (cdr s))) s) (else (let ((org_cddr (cddr s))) (set-cdr! (cdr s) (car s)) (set-car! s org_cddr) s))))
%Y A089852 a(n) = A069770(A089858(n)) = A089861(A069770(n)) = A057163(A089856(A057163(n))). Row 5 of A089840.
%Y A089852 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 A089852 nonn
%O A089852 0,3
%A A089852 _Antti Karttunen_, Nov 29 2003
%E A089852 Further comments and constructive implementation of Scheme-function (*A089852) added by _Antti Karttunen_, Jun 04 2011