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A089852
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Involution of natural numbers induced by Catalan automorphism *A089852 acting on the binary trees/parenthesizations encoded by A014486/A063171.
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14
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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
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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).
...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.
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LINKS
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PROG
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(Scheme functions implementing this automorphism on list-structures/S-expressions, both constructive (*A089852) and destructive (*A089852!) versions:)
(define (*A089852 s) (if (and (pair? s) (pair? (cdr s))) (cons (cddr s) (cons (cadr s) (car s))) s))
(define (*A089852! s) (cond ((not (pair? s)) s) ((not (pair? (cdr s))) s) (else (let ((org_cddr (cddr s))) (set-cdr! (cdr s) (car s)) (set-car! s org_cddr) s))))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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Further comments and constructive implementation of Scheme-function (*A089852) added by Antti Karttunen, Jun 04 2011
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STATUS
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approved
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