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%I #5 Jul 20 2018 22:30:16
%S 1,1,3,6,18,46,172,563,2347
%N Number of connected multiset partitions of strongly normal multisets of size n.
%C A multiset is strongly normal if it spans an initial interval of positive integers with weakly decreasing multiplicities.
%e The a(3) = 6 connected multiset partitions are (111), (1)(11), (1)(1)(1), (112), (1)(12), (123).
%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]]],multijoin@@s[[c[[1]]]]]]]]];
%t strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
%t Length/@Table[Join@@Table[Select[mps[m],Length[csm[#]]==1&],{m,strnorm[n]}],{n,8}]
%Y Cf. A007716, A007718, A048143, A293994, A303837, A303838, A304716, A305078.
%Y Cf. A317074, A317076, A317077, A317080.
%K nonn,more
%O 0,3
%A _Gus Wiseman_, Jul 20 2018
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