A089855 Permutation of natural numbers induced by Catalan Automorphism *A089855 acting on the binary trees/parenthesizations encoded by A014486/A063171.

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

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

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

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

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

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

In terms of S-expressions, this automorphism rotates caar, cdar and cdr of an S-exp if possible, i.e., if car-side 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 (*A089855) and destructive (*A089855!) versions:)
(define (*A089855 s) (if (and (pair? s) (pair? (car s))) (cons (cons (cdar s) (cdr s)) (caar s)) s))
(define (*A089855! s) (cond ((not (pair? s)) s) ((not (pair? (car s))) s) (else (let ((org_cdar (cdar s))) (set-cdr! (car s) (cdr s)) (set-cdr! s (caar s)) (set-car! (car s) org_cdar) s))))

Crossrefs

Inverse of A089857. a(n) = A089860(A069770(n)) = A069770(A074680(n)) = A057163(A089853(A057163(n))). Row 9 of A089840.

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

Sequence in context: A082352 A122320 A123713 * A122315 A130938 A130387

Adjacent sequences: A089852 A089853 A089854 * A089856 A089857 A089858

Keyword

nonn

Author

Antti Karttunen, Nov 29 2003

Extensions

Further comments and constructive implementation of Scheme-function (*A089855) added by Antti Karttunen, Jun 04 2011

Status

approved