A089860 Permutation of natural numbers induced by Catalan automorphism *A089860 acting on the binary trees/parenthesizations encoded by A014486/A063171.

0, 1, 3, 2, 8, 6, 7, 4, 5, 21, 22, 19, 14, 15, 20, 16, 17, 9, 10, 18, 11, 12, 13, 58, 59, 62, 63, 64, 56, 60, 51, 37, 38, 52, 39, 40, 41, 57, 61, 53, 42, 43, 54, 44, 45, 23, 24, 46, 25, 26, 27, 55, 47, 48, 28, 29, 49, 30, 31, 32, 50, 33, 34, 35, 36, 170, 171, 174, 175, 176, 184

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

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

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

(a . (b . c)) --> ((c . a) . b) _____ (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 (*A089860) and destructive (*A089860!) versions:)
(define (*A089860 s) (cond ((not (pair? s)) s) ((not (pair? (cdr s))) (cons (cdr s) (car s))) (else (cons (cons (cddr s) (car s)) (cadr s)))))
(define (*A089860! s) (cond ((pair? s) (cond ((pair? (cdr s)) (swap! (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 16 of A089840. Inverse of A089862. a(n) = A089855(A069770(n)) = A069770(A089851(n)) = A069770(A074680(A069770(n))) = A057163(A089862(A057163(n))).

Number of cycles: A001683 (probably, not checked). Number of fixed points: A019590. Max. cycle size & LCM of all cycle sizes: A089410 (in each range limited by A014137 and A014138).

Sequence in context: A260323 A072112 A071658 * A130960 A130927 A057162

Adjacent sequences: A089857 A089858 A089859 * A089861 A089862 A089863

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