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

0,3

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.

Links

Table of n, a(n) for n=0..11.

Example

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}}}}

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<Length[#]<Length[m]&]}], m];
Table[Sum[Length[Select[totm[m], Depth[#]==If[Length[m]<=1, 2, Length[m]]&]], {m, IntegerPartitions[n]}], {n, 0, 5}]

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.

Sequence in context: A208987 A176074 A281368 * A059801 A102226 A195530

Adjacent sequences: A330723 A330724 A330725 * A330727 A330728 A330729

Keyword

nonn,more

Author

Gus Wiseman, Jan 03 2020

Status

approved