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A331679
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Number of lone-child-avoiding locally disjoint rooted trees whose leaves are positive integers summing to n, with no two distinct leaves directly under the same vertex.
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12
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1, 2, 3, 8, 16, 48, 116, 341, 928, 2753, 7996, 24254, 73325, 226471, 702122
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OFFSET
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1,2
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COMMENTS
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A tree is locally disjoint if no child of any vertex has branches overlapping the branches of any other unequal child of the same vertex. It is lone-child-avoiding if there are no unary branchings.
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LINKS
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EXAMPLE
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The a(1) = 1 through a(5) = 16 trees:
1 2 3 4 5
(11) (111) (22) (11111)
(1(11)) (1111) ((11)3)
(2(11)) (1(22))
(1(111)) (2(111))
(11(11)) (1(1111))
((11)(11)) (11(111))
(1(1(11))) (111(11))
(1(2(11)))
(2(1(11)))
(1(1(111)))
(1(11)(11))
(1(11(11)))
(11(1(11)))
(1((11)(11)))
(1(1(1(11))))
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MATHEMATICA
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disjointQ[u_]:=Apply[And, Outer[#1==#2||Intersection[#1, #2]=={}&, u, u, 1], {0, 1}];
usot[n_]:=Prepend[Join@@Table[Select[Union[Sort/@Tuples[usot/@ptn]], disjointQ[DeleteCases[#, _?AtomQ]]&&SameQ@@Select[#, AtomQ]&], {ptn, Select[IntegerPartitions[n], Length[#]>1&]}], n];
Table[Length[usot[n]], {n, 12}]
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CROSSREFS
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The non-locally disjoint version is A141268.
Locally disjoint trees counted by vertices are A316473.
The case where all leaves are 1's is A316697.
Number of trees counted by A331678 with all atoms equal to 1.
Matula-Goebel numbers of locally disjoint rooted trees are A316495.
Unlabeled lone-child-avoiding locally disjoint rooted trees are A331680.
Cf. A000081, A000669, A001678, A005804, A060356, A300660, A316471, A316694, A316696, A319312, A330465, A331681.
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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