A089861 Permutation of natural numbers induced by Catalan Automorphism *A089861 acting on the binary trees/parenthesizations encoded by A014486/A063171.

0, 1, 3, 2, 7, 8, 4, 6, 5, 17, 18, 20, 21, 22, 9, 10, 14, 16, 19, 11, 15, 12, 13, 45, 46, 48, 49, 50, 54, 55, 57, 58, 59, 61, 62, 63, 64, 23, 24, 25, 26, 27, 37, 38, 42, 44, 47, 51, 53, 56, 60, 28, 29, 39, 43, 52, 30, 40, 31, 32, 33, 41, 34, 35, 36, 129, 130, 132, 133, 134, 138

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...............A...C

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

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

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

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

((a . b) . c) --> (b . (a . c)) ____ (() . 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..70.

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 (*A089861) and destructive (*A089861!) versions:)
(define (*A089861 s) (cond ((not (pair? s)) s) ((not (pair? (car s))) (cons (cdr s) (car s))) (else (cons (cdar s) (cons (caar s) (cdr s))))))
(define (*A089861! s) (cond ((pair? s) (cond ((pair? (car s)) (swap! (car s)) (robr! 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 18 of A089840. Inverse of A089858. a(n) = A089852(A069770(n)) = A069770(A072797(n)) = A057163(A073269(A057163(n))).

Number of cycles: A073193. Number of fixed-points: A019590. Max. cycle size: A089422. LCM of cycle sizes: A089423 (in each range limited by A014137 and A014138).

Sequence in context: A130998 A130926 A069774 * A072092 A071659 A130931

Adjacent sequences: A089858 A089859 A089860 * A089862 A089863 A089864

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