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A330628 Number of series/singleton-reduced rooted trees on strongly normal multisets of size n whose leaves are sets (not necessarily disjoint). 8
1, 1, 1, 5, 42, 423, 5458, 80926 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
A series/singleton-reduced rooted tree on a multiset m is either the multiset m itself or a sequence of series/singleton-reduced rooted trees, one on each part of a multiset partition of m that is neither minimal (all singletons) nor maximal (only one part).
A finite multiset is strongly normal if it covers an initial interval of positive integers with weakly decreasing multiplicities.
LINKS
EXAMPLE
The a(4) = 42 trees:
{{1}{1}{12}} {{12}{12}} {{1}{123}} {1234}
{{1}{{1}{12}}} {{1}{2}{12}} {{12}{13}} {{1}{234}}
{{1}{{2}{12}}} {{1}{1}{23}} {{12}{34}}
{{2}{{1}{12}}} {{1}{2}{13}} {{13}{24}}
{{1}{3}{12}} {{14}{23}}
{{1}{{1}{23}}} {{2}{134}}
{{1}{{2}{13}}} {{3}{124}}
{{1}{{3}{12}}} {{4}{123}}
{{2}{{1}{13}}} {{1}{2}{34}}
{{3}{{1}{12}}} {{1}{3}{24}}
{{1}{4}{23}}
{{2}{3}{14}}
{{2}{4}{13}}
{{3}{4}{12}}
{{1}{{2}{34}}}
{{1}{{3}{24}}}
{{1}{{4}{23}}}
{{2}{{1}{34}}}
{{2}{{3}{14}}}
{{2}{{4}{13}}}
{{3}{{1}{24}}}
{{3}{{2}{14}}}
{{3}{{4}{12}}}
{{4}{{1}{23}}}
{{4}{{2}{13}}}
{{4}{{3}{12}}}
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];
ssrtrees[m_]:=Prepend[Join@@Table[Tuples[ssrtrees/@p], {p, Select[mps[m], Length[m]>Length[#1]>1&]}], m];
Table[Sum[Length[Select[ssrtrees[s], FreeQ[#, {___, x_Integer, x_Integer, ___}]&]], {s, strnorm[n]}], {n, 0, 5}]
CROSSREFS
The generalization where leaves are multisets is A330471.
The non-singleton-reduced version is A330625.
The unlabeled version is A330626.
The case with all atoms distinct is A000311.
Strongly normal multiset partitions are A035310.
Sequence in context: A082145 A126765 A228793 * A327272 A024492 A217805
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Dec 26 2019
STATUS
approved

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Last modified March 29 04:23 EDT 2024. Contains 371264 sequences. (Running on oeis4.)