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%I #6 Nov 01 2018 11:37:35
%S 0,1,3,6,17,43,147,458,1729,6445,27011
%N Number of connected multiset partitions with multiset density -1 of strongly normal multisets of size n.
%C The multiset density of a multiset partition is the sum of the numbers of distinct vertices in each part minus the number of parts minus the number of vertices.
%C A multiset is normal if it spans an initial interval of positive integers, and strongly normal if in addition its multiplicities are weakly decreasing.
%e The a(1) = 1 through a(4) = 17 multiset partitions:
%e {{1}} {{1,1}} {{1,1,1}} {{1,1,1,1}}
%e {{1,2}} {{1,1,2}} {{1,1,1,2}}
%e {{1},{1}} {{1,2,3}} {{1,1,2,2}}
%e {{1},{1,1}} {{1,1,2,3}}
%e {{1},{1,2}} {{1,2,3,4}}
%e {{1},{1},{1}} {{1},{1,1,1}}
%e {{1,1},{1,1}}
%e {{1},{1,1,2}}
%e {{1,1},{1,2}}
%e {{1},{1,2,2}}
%e {{1},{1,2,3}}
%e {{1,2},{1,3}}
%e {{2},{1,1,2}}
%e {{1},{1},{1,1}}
%e {{1},{1},{1,2}}
%e {{1},{2},{1,2}}
%e {{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 csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Union[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
%t mensity[c_]:=Total[(Length[Union[#]]-1&)/@c]-Length[Union@@c];
%t strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
%t Table[Sum[Length[Select[mps[m],And[mensity[#]==-1,Length[csm[#]]==1]&]],{m,strnorm[n]}],{n,0,8}]
%Y Cf. A000272, A007716, A007718, A030019, A052888, A134954, A304867, A304887, A318697, A321155, A321228, A321229, A321231.
%K nonn,more
%O 0,3
%A _Gus Wiseman_, Oct 31 2018