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A089862 Permutation of natural numbers induced by Catalan automorphism *A089862 acting on the binary trees/parenthesizations encoded by A014486/A063171. 13
0, 1, 3, 2, 7, 8, 5, 6, 4, 17, 18, 20, 21, 22, 12, 13, 15, 16, 19, 11, 14, 9, 10, 45, 46, 48, 49, 50, 54, 55, 57, 58, 59, 61, 62, 63, 64, 31, 32, 34, 35, 36, 40, 41, 43, 44, 47, 52, 53, 56, 60, 30, 33, 39, 42, 51, 28, 37, 23, 24, 29, 38, 25, 26, 27, 129, 130, 132, 133, 134 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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).

.A...B...............C...A

..\./.................\./

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

....\./.............\./.................\./....-->....\./...

.....x...............x...................x.............x....

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

See the Karttunen OEIS-Wiki link for a detailed explanation of how to obtain a given integer sequence from this definition.

LINKS

Table of n, a(n) for n=0..69.

A. Karttunen, Catalan Automorphisms

A. Karttunen, C-program for computing this sequence

Index entries for signature-permutations induced by Catalan automorphisms

PROG

(Scheme functions implementing this automorphism on list-structures/S-expressions, both constructive (*A089862) and destructive (*A089862!) versions:)

(define (*A089862 s) (cond ((not (pair? s)) s) ((not (pair? (car s))) (cons (cdr s) (car s))) (else (cons (cdar s) (cons (cdr s) (caar s))))))

(define (*A089862! s) (cond ((pair? s) (cond ((pair? (car s)) (swap! (car s)) (robr! s) (swap! (cdr s))) (else (swap! s))))) s)

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

(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))

CROSSREFS

Row 20 of A089840. Inverse of A089860. a(n) = A089853(A069770(n)) = A069770(A089857(n)) = A069770(A074679(A069770(n))) = A057163(A089860(A057163(n))). Number of cycles: A001683 (seems to be, not checked). Number of fixed points: A019590. Max. cycle size & LCM of all cycle sizes: A089410 (in each range limited by A014137 and A014138).

Sequence in context: A071657 A057161 A130363 * A125981 A122326 A130996

Adjacent sequences:  A089859 A089860 A089861 * A089863 A089864 A089865

KEYWORD

nonn

AUTHOR

Antti Karttunen, Nov 29 2003

EXTENSIONS

A graphical description and constructive version of Scheme-implementation added by Antti Karttunen, Jun 04 2011

STATUS

approved

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Last modified March 18 12:10 EDT 2019. Contains 321283 sequences. (Running on oeis4.)