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
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Dec 25 2019
STATUS
approved