A330679 Number of balanced reduced multisystems whose atoms constitute an integer partition of n.

1, 1, 2, 4, 12, 40, 180, 936, 5820, 41288, 331748, 2968688, 29307780, 316273976, 3704154568, 46788812168, 634037127612, 9174782661984, 141197140912208, 2302765704401360, 39671953757409256, 719926077632193848, 13726066030661998220, 274313334040504957368

Offset: 0

Author

Gus Wiseman, Dec 31 2019

Keywords

• nonn

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.

Examples

			The a(0) = 1 through a(4) = 12 multisystems:
  {}  {1}  {2}    {3}          {4}
           {1,1}  {1,2}        {1,3}
                  {1,1,1}      {2,2}
                  {{1},{1,1}}  {1,1,2}
                               {1,1,1,1}
                               {{1},{1,2}}
                               {{2},{1,1}}
                               {{1},{1,1,1}}
                               {{1,1},{1,1}}
                               {{1},{1},{1,1}}
                               {{{1}},{{1},{1,1}}}
                               {{{1,1}},{{1},{1}}}
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..200

Crossrefs

The case where the atoms are all 1's is A318813 = a(n)/2.

The version where the atoms constitute a strongly normal multiset is A330475.

The version where the atoms cover an initial interval is A330655.

The maximum-depth version is A330726.

Cf. A000041, A000111, A000669, A001970, A002846, A005121, A141268, A196545, A213427, A318812, A320160, A330474.

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]]]];
totm[m_]:=Prepend[Join@@Table[totm[p],{p,Select[mps[m],1
				

Formula

a(n > 1) = 2 * A318813(n).

Extensions

a(12) onwards from Andrew Howroyd, Jan 20 2024