cp's OEIS Frontend

A330475 Number of balanced reduced multisystems whose atoms constitute a strongly normal multiset of size n.

%I A330475 #6 Dec 28 2019 17:03:49
%S A330475 1,1,2,9,85,1143,25270
%N A330475 Number of balanced reduced multisystems whose atoms constitute a strongly normal multiset of size n.
%C A330475 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 A330475 A finite multiset is strongly normal if it covers an initial interval of positive integers with weakly decreasing multiplicities.
%e A330475 The a(0) = 1 through a(3) = 9 multisystems:
%e A330475   {}  {1}  {1,1}  {1,1,1}
%e A330475            {1,2}  {1,1,2}
%e A330475                   {1,2,3}
%e A330475                   {{1},{1,1}}
%e A330475                   {{1},{1,2}}
%e A330475                   {{1},{2,3}}
%e A330475                   {{2},{1,1}}
%e A330475                   {{2},{1,3}}
%e A330475                   {{3},{1,2}}
%t A330475 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A330475 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t A330475 strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
%t A330475 totm[m_]:=Prepend[Join@@Table[totm[p],{p,Select[mps[m],1<Length[#]<Length[m]&]}],m];
%t A330475 Table[Sum[Length[totm[m]],{m,strnorm[n]}],{n,0,5}]
%Y A330475 The (weakly) normal version is A330655.
%Y A330475 The maximum-depth case is A330675.
%Y A330475 The case where the atoms are {1..n} is A005121.
%Y A330475 The case where the atoms are all 1's is A318813.
%Y A330475 The tree version is A330471.
%Y A330475 Multiset partitions of strongly normal multisets are A035310.
%Y A330475 Cf. A000311, A000669, A001678, A316652, A318812, A330467, A330474, A330628, A330663, A330679.
%K A330475 nonn,more
%O A330475 0,3
%A A330475 _Gus Wiseman_, Dec 27 2019