

A089852


Involution of natural numbers induced by Catalan automorphism *A089852 acting on the binary trees/parenthesizations encoded by A014486/A063171.


14



0, 1, 2, 3, 6, 5, 4, 7, 8, 16, 19, 15, 12, 13, 14, 11, 9, 17, 18, 10, 20, 21, 22, 44, 47, 53, 56, 60, 43, 52, 40, 31, 32, 41, 34, 35, 36, 42, 51, 39, 30, 33, 37, 28, 23, 45, 46, 24, 48, 49, 50, 38, 29, 25, 54, 55, 26, 57, 58, 59, 27, 61, 62, 63, 64, 128, 131, 137, 140, 144
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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...........B...A
....\./.............\./
.A...x....>....C...x.................A..().........A...()..
..\./.............\./...................\./....>....\./...
...x...............x.....................x.............x....
(a . (b . c)) > (c . (b . a)) ______ (a . ()) > (a . ())
In terms of Sexpressions, this automorphism swaps car and cddr of an Sexp if its length > 1, if possible, otherwise keeps it intact.
See the Karttunen OEISWiki 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, Cprogram for computing this sequence
Index entries for signaturepermutations induced by Catalan automorphisms


PROG

(Scheme functions implementing this automorphism on liststructures/Sexpressions, both constructive (*A089852) and destructive (*A089852!) versions:)
(define (*A089852 s) (if (and (pair? s) (pair? (cdr s))) (cons (cddr s) (cons (cadr s) (car s))) s))
(define (*A089852! s) (cond ((not (pair? s)) s) ((not (pair? (cdr s))) s) (else (let ((org_cddr (cddr s))) (setcdr! (cdr s) (car s)) (setcar! s org_cddr) s))))


CROSSREFS

a(n) = A069770(A089858(n)) = A089861(A069770(n)) = A057163(A089856(A057163(n))). Row 5 of A089840.
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).
Sequence in context: A130383 A130950 A122345 * A122308 A122307 A188568
Adjacent sequences: A089849 A089850 A089851 * A089853 A089854 A089855


KEYWORD

nonn


AUTHOR

Antti Karttunen, Nov 29 2003


EXTENSIONS

Further comments and constructive implementation of Schemefunction (*A089852) added by Antti Karttunen, Jun 04 2011


STATUS

approved



