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A324764
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Number of anti-transitive rooted identity trees with n nodes.
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21
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1, 1, 1, 1, 3, 4, 9, 20, 41, 89, 196, 443, 987, 2246, 5114, 11757, 27122, 62898, 146392, 342204, 802429, 1887882
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OFFSET
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1,5
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COMMENTS
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A rooted identity tree is an unlabeled rooted tree with no repeated branches directly under the same root. It is anti-transitive if the branches of the branches of the root are disjoint from the branches of the root.
Also the number of finitary sets S with n brackets where no element of an element of S is also an element of S. For example, the a(8) = 20 finitary sets are (o = {}):
{{{{{{{o}}}}}}}
{{{{{o,{o}}}}}}
{{{{o,{{o}}}}}}
{{{o,{{{o}}}}}}
{{{o,{o,{o}}}}}
{{{{o},{{o}}}}}
{{o,{{{{o}}}}}}
{{o,{{o,{o}}}}}
{{o,{o,{{o}}}}}
{{{o},{{{o}}}}}
{{{o},{o,{o}}}}
{{o,{o},{{o}}}}
{o,{{{{{o}}}}}}
{o,{{{o,{o}}}}}
{o,{{o,{{o}}}}}
{o,{{o},{{o}}}}
{{o},{{{{o}}}}}
{{o},{{o,{o}}}}
{{o},{o,{{o}}}}
{{{o}},{o,{o}}}
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LINKS
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EXAMPLE
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The a(1) = 1 through a(7) = 9 anti-transitive rooted identity trees:
o (o) ((o)) (((o))) ((o(o))) (((o(o)))) ((o(o(o))))
(o((o))) ((o((o)))) (o((o(o))))
((((o)))) (o(((o)))) ((((o(o)))))
(((((o))))) (((o)((o))))
(((o((o)))))
((o)(((o))))
((o(((o)))))
(o((((o)))))
((((((o))))))
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MATHEMATICA
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idall[n_]:=If[n==1, {{}}, Select[Union[Sort/@Join@@(Tuples[idall/@#]&/@IntegerPartitions[n-1])], UnsameQ@@#&]];
Table[Length[Select[idall[n], Intersection[Union@@#, #]=={}&]], {n, 10}]
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CROSSREFS
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Cf. A324694, A324751, A324756, A324758, A324765, A324767, A324768, A324770, A324839, A324840, A324844.
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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