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A316766
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Number of series-reduced locally stable rooted identity trees whose leaves form an integer partition of n.
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0
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1, 1, 2, 3, 6, 13, 30, 72, 180, 458, 1194, 3160, 8459, 22881, 62417, 171526, 474405, 1319395, 3687711, 10352696, 29178988
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OFFSET
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1,3
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COMMENTS
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A rooted tree is series-reduced if every non-leaf node has at least two branches. It is locally stable if no branch is a submultiset of any other branch of the same root. It is an identity tree if no branch appears multiple times under the same root.
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LINKS
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EXAMPLE
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The a(6) = 13 trees:
6,
(15),
(1(14)),
(1(1(13))),
(1(1(1(12)))),
(1(23)), (2(13)), (3(12)), (123),
(1(2(12))), (2(1(12))), (12(12)),
(24).
Example of non-stable trees are ((12)(123)) and ((12)(12(12))).
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MATHEMATICA
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submultisetQ[M_, N_]:=Or[Length[M]==0, MatchQ[{Sort[List@@M], Sort[List@@N]}, {{x_, Z___}, {___, x_, W___}}/; submultisetQ[{Z}, {W}]]];
stableQ[u_]:=Apply[And, Outer[#1==#2||!submultisetQ[#1, #2]&&!submultisetQ[#2, #1]&, u, u, 1], {0, 1}];
nms[n_]:=nms[n]=Prepend[Join@@Table[Select[Union[Sort/@Tuples[nms/@ptn]], And[UnsameQ@@#, stableQ[#]]&], {ptn, Rest[IntegerPartitions[n]]}], {n}];
Table[Length[nms[n]], {n, 10}]
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CROSSREFS
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Cf. A000081, A000669, A001678, A004111, A141268, A292504, A300660, A316467, A316474, A316653, A316654, A316656.
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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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