A089852 Involution of natural numbers induced by Catalan automorphism *A089852 acting on the binary trees/parenthesizations encoded by A014486/A063171.

0, 1, 2, 3, 6, 5, 4, 7, 8, 16, 19, 15, 12, 13, 14, 11, 9, 17, 18, 10, 20, 21, 22, 44, 47, 53, 56, 60, 43, 52, 40, 31, 32, 41, 34, 35, 36, 42, 51, 39, 30, 33, 37, 28, 23, 45, 46, 24, 48, 49, 50, 38, 29, 25, 54, 55, 26, 57, 58, 59, 27, 61, 62, 63, 64, 128, 131, 137, 140, 144

Offset: 0

Author

Antti Karttunen, Nov 29 2003

Keywords

• nonn

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

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

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

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

In terms of S-expressions, this automorphism swaps car and cddr of an S-exp if its length > 1, if possible, otherwise keeps it intact.

See the Karttunen OEIS-Wiki link for a detailed explanation of how to obtain a given integer sequence from this definition.

Links

• A. Karttunen, Catalan Automorphisms
• A. Karttunen, C-program for computing this sequence
• Index entries for signature-permutations induced by Catalan automorphisms

Crossrefs

a(n) = A069770(A089858(n)) = A089861(A069770(n)) = A057163(A089856(A057163(n))). Row 5 of A089840.

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).

Extensions

Further comments and constructive implementation of Scheme-function (*A089852) added by Antti Karttunen, Jun 04 2011