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

1, 1, 2, 9, 85, 1143, 25270

Offset: 0

Author

Gus Wiseman, Dec 27 2019

Keywords

• nonn
• more

Comments

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.

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

Examples

			The a(0) = 1 through a(3) = 9 multisystems:
  {}  {1}  {1,1}  {1,1,1}
           {1,2}  {1,1,2}
                  {1,2,3}
                  {{1},{1,1}}
                  {{1},{1,2}}
                  {{1},{2,3}}
                  {{2},{1,1}}
                  {{2},{1,3}}
                  {{3},{1,2}}
		

Crossrefs

The (weakly) normal version is A330655.

The maximum-depth case is A330675.

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

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

The tree version is A330471.

Multiset partitions of strongly normal multisets are A035310.

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

Programs

Mathematica

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
totm[m_]:=Prepend[Join@@Table[totm[p],{p,Select[mps[m],1