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A089853 Permutation of natural numbers induced by Catalan automorphism *A089853 acting on the binary trees/parenthesizations encoded by A014486/A063171. 17
0, 1, 2, 3, 6, 4, 5, 7, 8, 16, 19, 14, 9, 10, 15, 11, 12, 17, 18, 13, 20, 21, 22, 44, 47, 53, 56, 60, 42, 51, 37, 23, 24, 38, 25, 26, 27, 43, 52, 39, 28, 29, 40, 30, 31, 45, 46, 32, 48, 49, 50, 41, 33, 34, 54, 55, 35, 57, 58, 59, 36, 61, 62, 63, 64, 128, 131, 137, 140, 144 (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).

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

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

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

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

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

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

In terms of S-expressions, this automorphism rotates car, cadr and cddr of an S-exp if its length > 1.

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 (*A089853) and destructive (*A089853!) versions:)

(define (*A089853 s) (if (and (pair? s) (pair? (cdr s))) (cons (cddr s) (cons (car s) (cadr s))) s))

(define (*A089853! s) (cond ((not (pair? s)) s) ((not (pair? (cdr s))) s) (else (let ((org_cadr (cadr s))) (set-car! (cdr s) (car s)) (set-car! s (cddr s)) (set-cdr! (cdr s) org_cadr) s))))

CROSSREFS

Inverse of A089851. a(n) = A072796(A089850(n)) = A057163(A089855(A057163(n))). Row 6 of A089840.

Number of cycles: A089847. Number of fixed-points: A089848 (in each range limited by A014137 and A014138).

Sequence in context: A131147 A130984 A154124 * A122306 A122309 A130347

Adjacent sequences:  A089850 A089851 A089852 * A089854 A089855 A089856

KEYWORD

nonn

AUTHOR

Antti Karttunen, Nov 29 2003

EXTENSIONS

Further comments and constructive implementation of Scheme-function (*A089853) added by Antti Karttunen, Jun 04 2011

STATUS

approved

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Last modified November 19 00:12 EST 2019. Contains 329310 sequences. (Running on oeis4.)