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A316654
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Number of series-reduced rooted identity trees whose leaves span an initial interval of positive integers with multiplicities an integer partition of n.
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9
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1, 1, 5, 39, 387, 4960, 74088, 1312716, 26239484, 595023510, 14908285892, 412903136867, 12448252189622, 407804188400373, 14380454869464352, 544428684832123828, 21991444994187529639, 945234507638271696504, 43042162953650721470752, 2071216980365429970912347
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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 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(3) = 5 trees are (1(12)), (1(23)), (2(13)), (3(12)), (123).
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MATHEMATICA
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sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
gro[m_]:=If[Length[m]==1, m, Select[Union[Sort/@Join@@(Tuples[gro/@#]&/@Select[mps[m], Length[#]>1&])], UnsameQ@@#&]];
Table[Sum[Length[gro[m]], {m, Flatten[MapIndexed[Table[#2, {#1}]&, #]]&/@IntegerPartitions[n]}], {n, 5}]
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PROG
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(PARI) \\ See links in A339645 for combinatorial species functions.
cycleIndexSeries(n)={my(v=vector(n)); v[1]=sv(1); for(n=2, #v, v[n]=polcoef(sWeighT(x*Ser(v[1..n])), n)); x*Ser(v)}
StronglyNormalLabelingsSeq(cycleIndexSeries(12)) \\ Andrew Howroyd, Jan 22 2021
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CROSSREFS
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Cf. A000081, A000311, A000669, A001678, A004111, A005804, A141268, A181821, A292504, A300660, A304660.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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