A330726 Number of balanced reduced multisystems of maximum depth whose atoms are positive integers summing to n.

1, 1, 2, 3, 7, 17, 54, 199, 869, 4341, 24514, 154187

Offset: 0

Author

Gus Wiseman, Jan 03 2020

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.

Examples

			The a(1) = 1 through a(5) = 17 multisystems (commas elided):
  {1}  {2}   {3}        {4}               {5}
       {11}  {12}       {22}              {23}
             {{1}{11}}  {13}              {14}
                        {{1}{12}}         {{1}{13}}
                        {{2}{11}}         {{1}{22}}
                        {{{1}}{{1}{11}}}  {{2}{12}}
                        {{{11}}{{1}{1}}}  {{3}{11}}
                                          {{{1}}{{1}{12}}}
                                          {{{11}}{{1}{2}}}
                                          {{{1}}{{2}{11}}}
                                          {{{12}}{{1}{1}}}
                                          {{{2}}{{1}{11}}}
                                          {{{{1}}}{{{1}}{{1}{11}}}}
                                          {{{{1}}}{{{11}}{{1}{1}}}}
                                          {{{{1}{1}}}{{{1}}{{11}}}}
                                          {{{{1}{11}}}{{{1}}{{1}}}}
                                          {{{{11}}}{{{1}}{{1}{1}}}}
		

Crossrefs

The case with all atoms equal to 1 is A000111.

The non-maximal version is A330679.

A tree version is A320160.

Cf. A000669, A002846, A005121, A141268, A196545, A213427, A317145, A318813, A330663, A330665, A330675, A330676, A330728.

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