%I #5 Oct 11 2018 10:10:24

%S 1,1,2,4,8,17,30,61,110,207,381,711,1250

%N Number of strict T_0 multiset partitions of integer partitions of n.

%C The dual of a multiset partition has, for each vertex, one part consisting of the indices (or positions) of the parts containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}. The T_0 condition means the dual is strict.

%e The a(1) = 1 through a(5) = 17 multiset partitions:

%e {{1}} {{2}} {{3}} {{4}} {{5}}

%e {{1,1}} {{1,1,1}} {{2,2}} {{1,1,3}}

%e {{1},{2}} {{1,1,2}} {{1,2,2}}

%e {{1},{1,1}} {{1},{3}} {{1},{4}}

%e {{1,1,1,1}} {{2},{3}}

%e {{1},{1,2}} {{1,1,1,2}}

%e {{2},{1,1}} {{1},{1,3}}

%e {{1},{1,1,1}} {{1},{2,2}}

%e {{2},{1,2}}

%e {{3},{1,1}}

%e {{1,1,1,1,1}}

%e {{1},{1,1,2}}

%e {{1,1},{1,2}}

%e {{2},{1,1,1}}

%e {{1},{1,1,1,1}}

%e {{1,1},{1,1,1}}

%e {{1},{2},{1,1}}

%t sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];

%t mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];

%t dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];

%t Table[Length[Select[Join@@mps/@IntegerPartitions[n],And[UnsameQ@@#,UnsameQ@@dual[#]]&]],{n,8}]

%Y Cf. A001970, A047968, A050342, A089259, A141268, A261049, A289501, A305551, A319066, A319312, A320328, A320330.

%K nonn,more

%O 0,3

%A _Gus Wiseman_, Oct 11 2018