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Permutation of natural numbers induced by Catalan Automorphism *A089861 acting on the binary trees/parenthesizations encoded by A014486/A063171.
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%I #21 May 01 2018 02:56:24

%S 0,1,3,2,7,8,4,6,5,17,18,20,21,22,9,10,14,16,19,11,15,12,13,45,46,48,

%T 49,50,54,55,57,58,59,61,62,63,64,23,24,25,26,27,37,38,42,44,47,51,53,

%U 56,60,28,29,39,43,52,30,40,31,32,33,41,34,35,36,129,130,132,133,134,138

%N Permutation of natural numbers induced by Catalan Automorphism *A089861 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...............A...C

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

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

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

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

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

%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 (*A089861) and destructive (*A089861!) versions:)

%o (define (*A089861 s) (cond ((not (pair? s)) s) ((not (pair? (car s))) (cons (cdr s) (car s))) (else (cons (cdar s) (cons (caar s) (cdr s))))))

%o (define (*A089861! s) (cond ((pair? s) (cond ((pair? (car s)) (swap! (car s)) (robr! s)) (else (swap! s))))) s)

%o (define (swap! s) (let ((ex-car (car s))) (set-car! s (cdr s)) (set-cdr! s ex-car) s))

%o (define (robr! s) (let ((ex-cdr (cdr s))) (set-cdr! s (caar s)) (set-car! (car s) ex-cdr) (swap! (car s)) (swap! s) s))

%Y Row 18 of A089840. Inverse of A089858. a(n) = A089852(A069770(n)) = A069770(A072797(n)) = A057163(A073269(A057163(n))).

%Y Number of cycles: A073193. Number of fixed-points: A019590. Max. cycle size: A089422. LCM of cycle sizes: A089423 (in each range limited by A014137 and A014138).

%K nonn

%O 0,3

%A _Antti Karttunen_, Nov 29 2003

%E A graphical description and constructive version of Scheme-implementation added by _Antti Karttunen_, Jun 04 2011