cp's OEIS Frontend

A330625 Number of series-reduced rooted trees whose leaves are sets (not necessarily disjoint) with multiset union a strongly normal multiset of size n.

%I A330625 #6 Apr 30 2020 09:33:48
%S A330625 1,1,3,14,123,1330,19694
%N A330625 Number of series-reduced rooted trees whose leaves are sets (not necessarily disjoint) with multiset union a strongly normal multiset of size n.
%C A330625 A rooted tree is series-reduced if it has no unary branchings, so every non-leaf node covers at least two other nodes.
%C A330625 A finite multiset is strongly normal if it covers an initial interval of positive integers with weakly decreasing multiplicities.
%e A330625 The a(1) = 1 through a(3) = 14 trees:
%e A330625   {1}  {1,2}      {1,2,3}
%e A330625        {{1},{1}}  {{1},{1,2}}
%e A330625        {{1},{2}}  {{1},{2,3}}
%e A330625                   {{2},{1,3}}
%e A330625                   {{3},{1,2}}
%e A330625                   {{1},{1},{1}}
%e A330625                   {{1},{1},{2}}
%e A330625                   {{1},{2},{3}}
%e A330625                   {{1},{{1},{1}}}
%e A330625                   {{1},{{1},{2}}}
%e A330625                   {{1},{{2},{3}}}
%e A330625                   {{2},{{1},{1}}}
%e A330625                   {{2},{{1},{3}}}
%e A330625                   {{3},{{1},{2}}}
%t A330625 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A330625 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t A330625 strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
%t A330625 srtrees[m_]:=Prepend[Join@@Table[Tuples[srtrees/@p],{p,Select[mps[m],Length[#1]>1&]}],m];
%t A330625 Table[Sum[Length[Select[srtrees[s],FreeQ[#,{___,x_Integer,x_Integer,___}]&]],{s,strnorm[n]}],{n,0,5}]
%Y A330625 The generalization where the leaves are multisets is A330467.
%Y A330625 The singleton-reduced case is A330628.
%Y A330625 The unlabeled version is A330624.
%Y A330625 The case with all atoms distinct is A005804.
%Y A330625 The case with all atoms equal is A196545.
%Y A330625 The case where all leaves are singletons is A330471.
%Y A330625 Cf. A000669, A004111, A141268, A300660, A316652, A330469, A330475, A330626, A330668, A330675.
%K A330625 nonn,more
%O A330625 0,3
%A A330625 _Gus Wiseman_, Dec 25 2019