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A330625
Number of series-reduced rooted trees whose leaves are sets (not necessarily disjoint) with multiset union a strongly normal multiset of size n.
8
1, 1, 3, 14, 123, 1330, 19694
OFFSET
0,3
COMMENTS
A rooted tree is series-reduced if it has no unary branchings, so every non-leaf node covers at least two other nodes.
A finite multiset is strongly normal if it covers an initial interval of positive integers with weakly decreasing multiplicities.
EXAMPLE
The a(1) = 1 through a(3) = 14 trees:
{1} {1,2} {1,2,3}
{{1},{1}} {{1},{1,2}}
{{1},{2}} {{1},{2,3}}
{{2},{1,3}}
{{3},{1,2}}
{{1},{1},{1}}
{{1},{1},{2}}
{{1},{2},{3}}
{{1},{{1},{1}}}
{{1},{{1},{2}}}
{{1},{{2},{3}}}
{{2},{{1},{1}}}
{{2},{{1},{3}}}
{{3},{{1},{2}}}
MATHEMATICA
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]]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2, {#1}]&, #]]&/@IntegerPartitions[n];
srtrees[m_]:=Prepend[Join@@Table[Tuples[srtrees/@p], {p, Select[mps[m], Length[#1]>1&]}], m];
Table[Sum[Length[Select[srtrees[s], FreeQ[#, {___, x_Integer, x_Integer, ___}]&]], {s, strnorm[n]}], {n, 0, 5}]
CROSSREFS
The generalization where the leaves are multisets is A330467.
The singleton-reduced case is A330628.
The unlabeled version is A330624.
The case with all atoms distinct is A005804.
The case with all atoms equal is A196545.
The case where all leaves are singletons is A330471.
Sequence in context: A127850 A324147 A186772 * A061029 A096657 A365997
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Dec 25 2019
STATUS
approved