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A123503
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An involution of nonnegative integers: signature permutation of a nonrecursive Catalan automorphism, row 253 of table A089840.
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6
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0, 1, 2, 3, 4, 6, 5, 8, 7, 9, 10, 14, 16, 19, 11, 15, 12, 21, 22, 13, 20, 17, 18, 23, 24, 25, 26, 27, 37, 38, 42, 44, 47, 51, 53, 56, 60, 28, 29, 39, 43, 52, 30, 40, 31, 58, 59, 32, 62, 63, 64, 33, 41, 34, 57, 61, 35, 54, 45, 46, 36, 55, 48, 49, 50, 65, 66, 67, 68, 69, 70, 71
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OFFSET
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0,3
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COMMENTS
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This automorphism either swaps (if A057515(n) > 1) the first two toplevel elements (of a general plane tree, like *A072796 does) and otherwise (if n > 1, A057515(n)=1) swaps the sides of the left hand side subtree of the S-expression (when viewed as a binary tree, like *A089854 does). This is illustrated below, where letters A, B and C refer to arbitrary subtrees located on those nodes and () stands for an implied terminal node.
...B...C.............A...C............A...B...........B...A
....\./...............\./..............\./.............\./
.A...x.....-->.....B...x................x..()....-->....x..()
..\./...............\./..................\./.............\./
(a . (b . c)) --> (b . (a . c)) / ((a . b) . ()) --> ((b . a) . ())
This is the first multiclause automorphism in table A089840 which cannot be represented as a composition of two smaller nonrecursive automorphisms, the property which is also shared by *A123499 and *A123500.
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LINKS
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PROG
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(Scheme function, destructive implementation of this automorphism acting on S-expressions:) (define (*A123503! s) (cond ((null? s) s) ((pair? (cdr s)) (*A072796! s)) (else (*A089854! s))) 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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STATUS
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approved
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