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%I #7 Dec 31 2019 08:24:15
%S 1,1,2,6,43,440,7158,151414
%N Number of balanced reduced multisystems of maximum depth 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(2) = 2 and a(3) = 6 multisystems:
%e {1,1} {{1},{1,1}}
%e {1,2} {{1},{1,2}}
%e {{1},{2,3}}
%e {{2},{1,1}}
%e {{2},{1,3}}
%e {{3},{1,2}}
%e The a(4) = 43 multisystems (commas and outer brackets elided):
%e {{1}}{{1}{11}} {{1}}{{1}{12}} {{1}}{{1}{22}} {{1}}{{1}{23}} {{1}}{{2}{34}}
%e {{11}}{{1}{1}} {{11}}{{1}{2}} {{11}}{{2}{2}} {{11}}{{2}{3}} {{12}}{{3}{4}}
%e {{1}}{{2}{11}} {{1}}{{2}{12}} {{1}}{{2}{13}} {{1}}{{3}{24}}
%e {{12}}{{1}{1}} {{12}}{{1}{2}} {{12}}{{1}{3}} {{13}}{{2}{4}}
%e {{2}}{{1}{11}} {{2}}{{1}{12}} {{1}}{{3}{12}} {{1}}{{4}{23}}
%e {{2}}{{2}{11}} {{13}}{{1}{2}} {{14}}{{2}{3}}
%e {{22}}{{1}{1}} {{2}}{{1}{13}} {{2}}{{1}{34}}
%e {{2}}{{3}{11}} {{2}}{{3}{14}}
%e {{23}}{{1}{1}} {{23}}{{1}{4}}
%e {{3}}{{1}{12}} {{2}}{{4}{13}}
%e {{3}}{{2}{11}} {{24}}{{1}{3}}
%e {{3}}{{1}{24}}
%e {{3}}{{2}{14}}
%e {{3}}{{4}{12}}
%e {{34}}{{1}{2}}
%e {{4}}{{1}{23}}
%e {{4}}{{2}{13}}
%e {{4}}{{3}{12}}
%t strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
%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 totm[m_]:=Prepend[Join@@Table[totm[p],{p,Select[mps[m],1<Length[#]<Length[m]&]}],m];
%t Table[Sum[Length[Select[totm[m],Depth[#]==If[Length[m]<=1,2,Length[m]]&]],{m,strnorm[n]}],{n,0,5}]
%Y The case with all atoms equal is A000111.
%Y The case with all atoms different is A006472.
%Y The version allowing all depths is A330475.
%Y The unlabeled version is A330663.
%Y The version where the atoms are the prime indices of n is A330665.
%Y The (weakly) normal version is A330676.
%Y The version where the degrees are the prime indices of n is A330728.
%Y Multiset partitions of strongly normal multisets are A035310.
%Y Series-reduced rooted trees with strongly normal leaves are A316652.
%Y Cf. A000311, A000669, A001055, A001678, A005121, A005804, A316651, A318812, A330467, A330474, A330625, A330628, A330664, A330677, A330679.
%K nonn,more
%O 0,3
%A _Gus Wiseman_, Dec 30 2019
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