A089850 Involution of natural numbers induced by Catalan automorphism *A089850 acting on the binary trees/parenthesizations encoded by A014486/A063171.

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

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

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

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

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

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

In terms of S-expressions, this automorphism swaps cadr and cddr of an S-exp if its length > 1.

Look at the example in A069770 to see how this will produce the given sequence of integers.

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, constructive (*A089850) and destructive (*A089850!) version:)
(define (*A089850 s) (if (and (pair? s) (pair? (cdr s))) (cons (car s) (cons (cddr s) (cadr s))) s))
(define (*A089850! s) (cond ((not (pair? s)) s) ((not (pair? (cdr s))) s) (else (*A069770! (cdr s)) s)))

Crossrefs

a(n) = A069770(A089859(n)) = A089863(A069770(n)) = A057163(A089854(A057163(n))). Row 3 of A089840. Row 3771 of A122203 and row 3677 of A122204.

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: A154121 A130375 A154122 * A154450 A154449 A122304

Adjacent sequences: A089847 A089848 A089849 * A089851 A089852 A089853

Keyword

nonn

Author

Antti Karttunen, Nov 29 2003

Extensions

The new mail-address, a graphical explanation and constructive implementation of Scheme-function (*A089850) added by Antti Karttunen, Jun 04 2011

Status

approved