A089851 Permutation of natural numbers induced by Catalan automorphism *A089851 acting on the binary trees/parenthesizations encoded by A014486/A063171.

0, 1, 2, 3, 5, 6, 4, 7, 8, 12, 13, 15, 16, 19, 11, 14, 9, 17, 18, 10, 20, 21, 22, 31, 32, 34, 35, 36, 40, 41, 43, 44, 47, 52, 53, 56, 60, 30, 33, 39, 42, 51, 28, 37, 23, 45, 46, 24, 48, 49, 50, 29, 38, 25, 54, 55, 26, 57, 58, 59, 27, 61, 62, 63, 64, 87, 88, 90, 91, 92, 96, 97, 99

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

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

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

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

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

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

In terms of S-expressions, this rotates car, cadr and cddr of an S-exp

if its length > 1, otherwise keeps it intact.

Note that the first clause corresponds to generator C of Thompson's groups T and V.

(Cf. also A072796, A074679 and A154121 for other related generators).

See "Catalan Automorphisms" OEIS-Wiki page for a detailed explanation 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, C-program for computing this sequence

Index entries for signature-permutations induced by Catalan automorphisms

Prog

(Scheme functions implementing this automorphism on list-structures, both constructive (*A089851) and destructive (*A089851!) versions:)
(define (*A089851 s) (if (and (pair? s) (pair? (cdr s))) (cons (cadr s) (cons (cddr s) (car s))) s))
(define (*A089851! s) (cond ((not (pair? s)) s) ((not (pair? (cdr s))) s) (else (swap! s) (robr! 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

Inverse of A089853. a(n) = A089850(A072796(n)) = A057163(A089857(A057163(n))). Row 4 of A089840.

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

Sequence in context: A130381 A130948 A122343 * A122310 A122305 A131143

Adjacent sequences: A089848 A089849 A089850 * A089852 A089853 A089854

Keyword

nonn

Author

Antti Karttunen, Nov 29 2003

Extensions

The new mail-address, further comments and constructive implementation of Scheme-function (*A089851) added by Antti Karttunen, Jun 04 2011

Status

approved