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A358458
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Numbers k such that the k-th standard ordered rooted tree is weakly transitive (counted by A358454).
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2
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1, 2, 4, 6, 7, 8, 12, 14, 15, 16, 18, 22, 23, 24, 25, 27, 28, 30, 31, 32, 36, 38, 39, 42, 44, 45, 46, 47, 48, 50, 51, 53, 54, 55, 56, 57, 59, 60, 62, 63, 64, 70, 71, 72, 76, 78, 79, 82, 84, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 99, 100, 102, 103, 105
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OFFSET
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1,2
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COMMENTS
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We define an unlabeled ordered rooted tree to be weakly transitive if every branch of a branch of the root is itself a branch of the root.
We define the n-th standard ordered rooted tree to be obtained by taking the (n-1)-th composition in standard order (graded reverse-lexicographic, A066099) as root and replacing each part with its own standard ordered rooted tree. This ranking is an ordered variation of Matula-Goebel numbers, giving a bijective correspondence between positive integers and unlabeled ordered rooted trees.
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LINKS
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Table of n, a(n) for n=1..65.
Gus Wiseman, Statistics, classes, and transformations of standard compositions
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EXAMPLE
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The terms together with their corresponding ordered trees begin:
1: o
2: (o)
4: (oo)
6: ((o)o)
7: (o(o))
8: (ooo)
12: ((o)oo)
14: (o(o)o)
15: (oo(o))
16: (oooo)
18: ((oo)o)
22: ((o)(o)o)
23: ((o)o(o))
24: ((o)ooo)
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MATHEMATICA
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stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
srt[n_]:=If[n==1, {}, srt/@stc[n-1]];
Select[Range[100], Complement[Union@@srt[#], srt[#]]=={}&]
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CROSSREFS
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The unordered version is A290822, counted by A290689.
These trees are counted by A358454.
The directed version is A358457, counted by A358453.
A000108 counts ordered rooted trees, unordered A000081.
A306844 counts anti-transitive rooted trees.
A324766 ranks recursively anti-transitive rooted trees, counted by A324765.
A358455 counts recursively anti-transitive ordered rooted trees.
Cf. A004249, A032027, A318185, A324695, A324758, A324766, A324840, A358373-A358377, A358456.
Sequence in context: A029453 A330943 A352089 * A014855 A298479 A015924
Adjacent sequences: A358455 A358456 A358457 * A358459 A358460 A358461
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KEYWORD
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nonn
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AUTHOR
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Gus Wiseman, Nov 18 2022
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STATUS
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approved
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