A089854 Involution of natural numbers induced by Catalan automorphism *A089854 acting on the binary trees/parenthesizations encoded by A014486/A063171.

0, 1, 2, 3, 4, 5, 6, 8, 7, 9, 10, 11, 12, 13, 14, 15, 19, 21, 22, 16, 20, 17, 18, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 51, 52, 56, 58, 59, 60, 62, 63, 64, 42, 43, 53, 57, 61, 44, 54, 45, 46, 47, 55, 48, 49, 50, 65, 66, 67, 68, 69, 70, 71, 72

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...........B...A

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

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

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

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

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

In terms of S-expressions, this automorphism swaps caar and cdar of an S-exp if its first element is not ().

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..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/S-expressions, both constructive (*A089854) and destructive (*A089854!) versions:)
(define (*A089854 s) (if (and (pair? s) (pair? (car s))) (cons (cons (cdar s) (caar s)) (cdr s)) s))
(define (*A089854! s) (cond ((not (pair? s)) s) ((not (pair? (car s))) s) (else (*A069770! (car s)) s)))

Crossrefs

a(n) = A089859(A069770(n)) = A069770(A089863(n)) = A057163(A089850(A057163(n))). Row 7 of A089840. Cf. A122282.

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: A353824 A089865 A089866 * A154453 A154454 A131173

Adjacent sequences: A089851 A089852 A089853 * A089855 A089856 A089857

Keyword

nonn

Author

Antti Karttunen, Nov 29 2003

Extensions

Further comments and constructive implementation of Scheme-function (*A089854) added by Antti Karttunen, Jun 04 2011

Status

approved