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Involution of natural numbers induced by Catalan automorphism *A089854 acting on the binary trees/parenthesizations encoded by A014486/A063171.
18

%I #24 May 01 2018 02:56:10

%S 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 26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,51,52,56,58,59,60,62,

%U 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 Involution of natural numbers induced by Catalan automorphism *A089854 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...........B...A

%C ..\./.............\./

%C ...x...C....-->....x...C...............()..A.........()..A..

%C ....\./.............\./.................\./....-->....\./...

%C .....x...............x...................x.............x....

%C ((a . b) . c) -> ((b . a) . c) _____ (() . a) ---> (() . a)

%C In terms of S-expressions, this automorphism swaps caar and cdar of an S-exp if its first element is not ().

%C See the Karttunen OEIS-Wiki 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">C-program for computing this sequence</a>

%H <a href="/index/Per#IntegerPermutationCatAuto">Index entries for signature-permutations induced by Catalan automorphisms</a>

%o (Scheme functions implementing this automorphism on list-structures/S-expressions, both constructive (*A089854) and destructive (*A089854!) versions:)

%o (define (*A089854 s) (if (and (pair? s) (pair? (car s))) (cons (cons (cdar s) (caar s)) (cdr s)) s))

%o (define (*A089854! s) (cond ((not (pair? s)) s) ((not (pair? (car s))) s) (else (*A069770! (car s)) s)))

%Y a(n) = A089859(A069770(n)) = A069770(A089863(n)) = A057163(A089850(A057163(n))). Row 7 of A089840. Cf. A122282.

%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 Scheme-function (*A089854) added by _Antti Karttunen_, Jun 04 2011