cp's OEIS Frontend

A089854 Involution of natural numbers induced by Catalan automorphism *A089854 acting on the binary trees/parenthesizations encoded by A014486/A063171.

%I A089854 #24 May 01 2018 02:56:10
%S A089854 0,1,2,3,4,5,6,8,7,9,10,11,12,13,14,15,19,21,22,16,20,17,18,23,24,25,
%T A089854 26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,51,52,56,58,59,60,62,
%U A089854 63,64,42,43,53,57,61,44,54,45,46,47,55,48,49,50,65,66,67,68,69,70,71,72
%N A089854 Involution of natural numbers induced by Catalan automorphism *A089854 acting on the binary trees/parenthesizations encoded by A014486/A063171.
%C A089854 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 A089854 .A...B...........B...A
%C A089854 ..\./.............\./
%C A089854 ...x...C....-->....x...C...............()..A.........()..A..
%C A089854 ....\./.............\./.................\./....-->....\./...
%C A089854 .....x...............x...................x.............x....
%C A089854 ((a . b) . c) -> ((b . a) . c) _____ (() . a) ---> (() . a)
%C A089854 In terms of S-expressions, this automorphism swaps caar and cdar of an S-exp if its first element is not ().
%C A089854 See the Karttunen OEIS-Wiki link for a detailed explanation of how to obtain a given integer sequence from this definition.
%H A089854 A. Karttunen, <a href="http://oeis.org/wiki/Catalan_Automorphisms">Catalan Automorphisms</a>
%H A089854 A. Karttunen, <a href="/A089408/a089408.c.txt">C-program for computing this sequence</a>
%H A089854 <a href="/index/Per#IntegerPermutationCatAuto">Index entries for signature-permutations induced by Catalan automorphisms</a>
%o A089854 (Scheme functions implementing this automorphism on list-structures/S-expressions, both constructive (*A089854) and destructive (*A089854!) versions:)
%o A089854 (define (*A089854 s) (if (and (pair? s) (pair? (car s))) (cons (cons (cdar s) (caar s)) (cdr s)) s))
%o A089854 (define (*A089854! s) (cond ((not (pair? s)) s) ((not (pair? (car s))) s) (else (*A069770! (car s)) s)))
%Y A089854 a(n) = A089859(A069770(n)) = A069770(A089863(n)) = A057163(A089850(A057163(n))). Row 7 of A089840. Cf. A122282.
%Y A089854 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 A089854 nonn
%O A089854 0,3
%A A089854 _Antti Karttunen_, Nov 29 2003
%E A089854 Further comments and constructive implementation of Scheme-function (*A089854) added by _Antti Karttunen_, Jun 04 2011