%I #6 Dec 28 2019 17:03:49

%S 1,1,2,9,85,1143,25270

%N Number of balanced reduced multisystems whose atoms constitute a strongly normal multiset of size n.

%C A balanced reduced multisystem is either a finite multiset, or a multiset partition with at least two parts, not all of which are singletons, of a balanced reduced multisystem.

%C A finite multiset is strongly normal if it covers an initial interval of positive integers with weakly decreasing multiplicities.

%e The a(0) = 1 through a(3) = 9 multisystems:

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

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

%e {1,2,3}

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

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

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

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

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

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

%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 strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];

%t totm[m_]:=Prepend[Join@@Table[totm[p],{p,Select[mps[m],1<Length[#]<Length[m]&]}],m];

%t Table[Sum[Length[totm[m]],{m,strnorm[n]}],{n,0,5}]

%Y The (weakly) normal version is A330655.

%Y The maximum-depth case is A330675.

%Y The case where the atoms are {1..n} is A005121.

%Y The case where the atoms are all 1's is A318813.

%Y The tree version is A330471.

%Y Multiset partitions of strongly normal multisets are A035310.

%Y Cf. A000311, A000669, A001678, A316652, A318812, A330467, A330474, A330628, A330663, A330679.

%K nonn,more

%O 0,3

%A _Gus Wiseman_, Dec 27 2019