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%I #6 Oct 11 2018 10:10:17
%S 1,1,3,5,13,25,50,100,195,366,707,1333,2440
%N Number of 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) = 25 multiset partitions:
%e {{1}} {{2}} {{3}} {{4}} {{5}}
%e {{1,1}} {{1,1,1}} {{2,2}} {{1,1,3}}
%e {{1},{1}} {{1},{2}} {{1,1,2}} {{1,2,2}}
%e {{1},{1,1}} {{1},{3}} {{1},{4}}
%e {{1},{1},{1}} {{2},{2}} {{2},{3}}
%e {{1,1,1,1}} {{1,1,1,2}}
%e {{1},{1,2}} {{1},{1,3}}
%e {{2},{1,1}} {{1},{2,2}}
%e {{1},{1,1,1}} {{2},{1,2}}
%e {{1,1},{1,1}} {{3},{1,1}}
%e {{1},{1},{2}} {{1,1,1,1,1}}
%e {{1},{1},{1,1}} {{1},{1,1,2}}
%e {{1},{1},{1},{1}} {{1,1},{1,2}}
%e {{1},{1},{3}}
%e {{1},{2},{2}}
%e {{2},{1,1,1}}
%e {{1},{1,1,1,1}}
%e {{1,1},{1,1,1}}
%e {{1},{1},{1,2}}
%e {{1},{2},{1,1}}
%e {{1},{1},{1,1,1}}
%e {{1},{1,1},{1,1}}
%e {{1},{1},{1},{2}}
%e {{1},{1},{1},{1,1}}
%e {{1},{1},{1},{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],UnsameQ@@dual[#]&]],{n,8}]
%Y Cf. A001970, A047968, A050342, A089259, A141268, A261049, A289501, A305551, A316983, A319066, A319312, A320328, A320331.
%K nonn,more
%O 0,3
%A _Gus Wiseman_, Oct 11 2018
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