A089859 Permutation of natural numbers induced by Catalan Automorphism *A089859 acting on the binary trees/parenthesizations encoded by A014486/A063171.

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

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

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

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

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

(a . (b . c)) --> ((c . b) . a) _____ (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 (*A089859) and destructive (*A089859!) versions:)
(define (*A089859 s) (cond ((not (pair? s)) s) ((not (pair? (cdr s))) (cons (cdr s) (car s))) (else (cons (cons (cddr s) (cadr s)) (car s)))))
(define (*A089859! s) (cond ((pair? s) (cond ((pair? (cdr s)) (*A069770! (cdr s)) (*A069770! s)) (else (*A069770! s))))) s)

Crossrefs

Row 15 of A089840. Inverse of A089863. a(n) = A089854(A069770(n)) = A069770(A089850(n)). A089864 is the "square" of this permutation.

Number of cycles: A089407. Max. cycle size & LCM of all cycle sizes: A040002 (in each range limited by A014137 and A014138).

Sequence in context: A122337 A125985 A073282 * A130958 A130397 A154457

Adjacent sequences: A089856 A089857 A089858 * A089860 A089861 A089862

Keyword

nonn

Author

Antti Karttunen, Nov 29 2003

Extensions

A graphical description and constructive implementation of Scheme-function (*A089859) added by Antti Karttunen, Jun 04 2011

Status

approved