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A089860 Permutation of natural numbers induced by Catalan automorphism *A089860 acting on the binary trees/parenthesizations encoded by A014486/A063171. 13

%I

%S 0,1,3,2,8,6,7,4,5,21,22,19,14,15,20,16,17,9,10,18,11,12,13,58,59,62,

%T 63,64,56,60,51,37,38,52,39,40,41,57,61,53,42,43,54,44,45,23,24,46,25,

%U 26,27,55,47,48,28,29,49,30,31,32,50,33,34,35,36,170,171,174,175,176,184

%N Permutation of natural numbers induced by Catalan automorphism *A089860 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 .....B...C.......C...A

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

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

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

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

%C (a . (b . c)) --> ((c . a) . b) _____ (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 (*A089860) and destructive (*A089860!) versions:)

%o (define (*A089860 s) (cond ((not (pair? s)) s) ((not (pair? (cdr s))) (cons (cdr s) (car s))) (else (cons (cons (cddr s) (car s)) (cadr s)))))

%o (define (*A089860! s) (cond ((pair? s) (cond ((pair? (cdr s)) (swap! (cdr s)) (robl! s) (swap! (car 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 (robl! s) (let ((ex-car (car s))) (set-car! s (cddr s)) (set-cdr! (cdr s) ex-car) (swap! (cdr s)) (swap! s) s))

%Y Row 16 of A089840. Inverse of A089862. a(n) = A089855(A069770(n)) = A069770(A089851(n)) = A069770(A074680(A069770(n))) = A057163(A089862(A057163(n))).

%Y Number of cycles: A001683 (probably, not checked). Number of fixed points: A019590. Max. cycle size & LCM of all cycle sizes: A089410 (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

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Last modified January 27 15:14 EST 2020. Contains 331295 sequences. (Running on oeis4.)