A089858 Permutation of natural numbers induced by Catalan Automorphism *A089858 acting on the binary trees/parenthesizations encoded by A014486/A063171.

0, 1, 3, 2, 6, 8, 7, 4, 5, 14, 15, 19, 21, 22, 16, 20, 17, 9, 10, 18, 11, 12, 13, 37, 38, 39, 40, 41, 51, 52, 56, 58, 59, 60, 62, 63, 64, 42, 43, 53, 57, 61, 44, 54, 45, 23, 24, 46, 25, 26, 27, 47, 55, 48, 28, 29, 49, 30, 31, 32, 50, 33, 34, 35, 36, 107, 108, 109, 110, 111

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

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

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

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

See the Karttunen OEIS-Wiki link for a detailed explanation 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 (*A089858) and destructive (*A089858!) versions:)
(define (*A089858 s) (cond ((not (pair? s)) s) ((not (pair? (cdr s))) (cons (cdr s) (car s))) (else (cons (cons (cadr s) (car s)) (cddr s)))))
(define (*A089858! s) (cond ((pair? s) (cond ((pair? (cdr s)) (robl! s) (swap! (car 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 (robl! s) (let ((ex-car (car s))) (set-car! s (cddr s)) (set-cdr! (cdr s) ex-car) (swap! (cdr s)) (swap! s) s))

Crossrefs

Row 13 of A089840. Inverse of A089861. a(n) = A072797(A069770(n)) = A069770(A089852(n)) = A057163(A073270(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: A122321 A072093 A130925 * A071660 A130393 A130956

Adjacent sequences: A089855 A089856 A089857 * A089859 A089860 A089861

Keyword

nonn

Author

Antti Karttunen, Nov 29 2003

Extensions

Further comments and constructive implementation of Scheme-function (*A089858) added by Antti Karttunen, Jun 04 2011

Status

approved