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A089855
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Permutation of natural numbers induced by Catalan Automorphism *A089855 acting on the binary trees/parenthesizations encoded by A014486/A063171.
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14
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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
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OFFSET
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0,3
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COMMENTS
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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 page <a href="http://oeis.org/wiki/Catalan_Automorphisms">Catalan Automorphisms</a> in OEIS-Wiki for a detailed explanation how to obtain a given integer sequence from this definition.
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LINKS
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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
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PROG
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(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))))
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CROSSREFS
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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
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KEYWORD
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nonn
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AUTHOR
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Antti Karttunen, Nov 29 2003
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EXTENSIONS
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Further comments and constructive implementation of Scheme-function (*A089855) added by Antti Karttunen, Jun 04 2011
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STATUS
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approved
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