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A330726 Number of balanced reduced multisystems of maximum depth whose atoms are positive integers summing to n. 2
1, 1, 2, 3, 7, 17, 54, 199, 869, 4341, 24514, 154187 (list; graph; refs; listen; history; text; internal format)
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

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Last modified July 27 15:49 EDT 2021. Contains 346308 sequences. (Running on oeis4.)