

A089856


Involution of natural numbers induced by Catalan Automorphism *A089856 acting on the binary trees/parenthesizations encoded by A014486/A063171.


12



0, 1, 2, 3, 4, 5, 8, 7, 6, 9, 10, 11, 12, 13, 21, 22, 20, 17, 18, 19, 16, 14, 15, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 58, 59, 62, 63, 64, 57, 61, 54, 45, 46, 55, 48, 49, 50, 56, 60, 53, 44, 47, 51, 42, 37, 38, 52, 43, 39, 40, 41, 65, 66, 67, 68, 69, 70, 71, 72
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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).
.A...B...........C...B
..\./.............\./
...x...C....>....x...A...............()..A.........()..A..
....\./.............\./.................\./....>....\./...
.....x...............x...................x.............x....
((a . b) . c) > ((c . b) . a) _____ (() . a) > (() . a)
In terms of Sexpressions, this automorphism swaps caar and cdr of an Sexp if possible, i.e., if carside is not ().
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..72.
A. Karttunen, Catalan Automorphisms
A. Karttunen, Cprogram for computing this sequence
Index entries for signaturepermutations induced by Catalan automorphisms


PROG

(Scheme function implementing this automorphism on liststructures/Sexpressions, both constructive (*A089856) and destructive (*A089856!) versions:)
(define (*A089856 s) (if (and (pair? s) (pair? (car s))) (cons (cons (cdr s) (cdar s)) (caar s)) s))
(define (*A089856! s) (cond ((not (pair? s)) s) ((not (pair? (car s))) s) (else (let ((org_caar (caar s))) (setcar! (car s) (cdr s)) (setcdr! s org_caar) s))))


CROSSREFS

Row 10 of A089840. a(n) = A073269(A069770(n)) = A069770(A073270(n)) = A057163(A089852(A057163(n))).
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: A130992 A122318 A130991 * A131151 A131152 A130939
Adjacent sequences: A089853 A089854 A089855 * A089857 A089858 A089859


KEYWORD

nonn


AUTHOR

Antti Karttunen, Nov 29 2003


EXTENSIONS

Further comments and constructive implementation of Schemefunction (*A089856) added by Antti Karttunen, Jun 04 2011


STATUS

approved



