A330668 -id:A330668 - cp's OEIS frontend

A330474 Number of non-isomorphic balanced reduced multisystems of weight n.

1, 1, 2, 7, 48, 424

Offset: 0

Gus Wiseman, Dec 26 2019

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. The weight of an atom is 1, while the weight of a multiset is the sum of weights of its elements.

			Non-isomorphic representatives of the a(3) = 7 multisystems:
  {1,1,1}
  {1,1,2}
  {1,2,3}
  {{1},{1,1}}
  {{1},{1,2}}
  {{1},{2,3}}
  {{2},{1,1}}
Non-isomorphic representatives of the a(4) = 48 multisystems:
  {1,1,1,1}  {{1},{1,1,1}}    {{{1}},{{1},{1,1}}}
  {1,1,1,2}  {{1,1},{1,1}}    {{{1,1}},{{1},{1}}}
  {1,1,2,2}  {{1},{1,1,2}}    {{{1}},{{1},{1,2}}}
  {1,1,2,3}  {{1,1},{1,2}}    {{{1,1}},{{1},{2}}}
  {1,2,3,4}  {{1},{1,2,2}}    {{{1}},{{1},{2,2}}}
             {{1,1},{2,2}}    {{{1,1}},{{2},{2}}}
             {{1},{1,2,3}}    {{{1}},{{1},{2,3}}}
             {{1,1},{2,3}}    {{{1,1}},{{2},{3}}}
             {{1,2},{1,2}}    {{{1}},{{2},{1,1}}}
             {{1,2},{1,3}}    {{{1,2}},{{1},{1}}}
             {{1},{2,3,4}}    {{{1}},{{2},{1,2}}}
             {{1,2},{3,4}}    {{{1,2}},{{1},{2}}}
             {{2},{1,1,1}}    {{{1}},{{2},{1,3}}}
             {{2},{1,1,3}}    {{{1,2}},{{1},{3}}}
             {{1},{1},{1,1}}  {{{1}},{{2},{3,4}}}
             {{1},{1},{1,2}}  {{{1,2}},{{3},{4}}}
             {{1},{1},{2,2}}  {{{2}},{{1},{1,1}}}
             {{1},{1},{2,3}}  {{{2}},{{1},{1,3}}}
             {{1},{2},{1,1}}  {{{2}},{{3},{1,1}}}
             {{1},{2},{1,2}}  {{{2,3}},{{1},{1}}}
             {{1},{2},{1,3}}
             {{1},{2},{3,4}}
             {{2},{3},{1,1}}

Labeled versions are A330475 (strongly normal) and A330655 (normal).
The case where the atoms are all different is A318813.
The case where the atoms are all equal is (also) A318813.
The labeled case of set partitions is A005121.
The labeled case of integer partitions is A330679.
The case of maximal depth is A330663.
The version where leaves are sets (as opposed to multisets) is A330668.
Cf. A000311, A000669, A001678, A002846, A004114, A007716, A048816, A213427, A306186, A320154, A320160, A330470, A330666.

A330624 Number of non-isomorphic series-reduced rooted trees whose leaves are sets (not necessarily disjoint) with a total of n elements.

Original entry on oeis.org

1, 1, 3, 10, 61, 410, 3630

Offset: 0

Gus Wiseman, Dec 25 2019

A rooted tree is series-reduced if it has no unary branchings, so every non-leaf node covers at least two other nodes.

			Non-isomorphic representatives of the a(1) = 1 through a(3) = 10 trees:
  {1}  {1,2}      {1,2,3}
       {{1},{1}}  {{1},{1,2}}
       {{1},{2}}  {{1},{2,3}}
                  {{1},{1},{1}}
                  {{1},{1},{2}}
                  {{1},{2},{3}}
                  {{1},{{1},{1}}}
                  {{1},{{1},{2}}}
                  {{1},{{2},{3}}}
                  {{2},{{1},{1}}}

The version with multisets as leaves is A330465.
The singleton-reduced case is A330626.
A labeled version is A330625 (strongly normal).
The case with all atoms distinct is A141268.
The case where all leaves are singletons is A330470.
Cf. A000669, A004111, A005804, A007716, A273873, A300660, A320296, A330628, A330668, A330677.

A330625 Number of series-reduced rooted trees whose leaves are sets (not necessarily disjoint) with multiset union a strongly normal multiset of size n.

Original entry on oeis.org

1, 1, 3, 14, 123, 1330, 19694

Offset: 0

Gus Wiseman, Dec 25 2019

A rooted tree is series-reduced if it has no unary branchings, so every non-leaf node covers at least two other nodes.

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

			The a(1) = 1 through a(3) = 14 trees:
  {1}  {1,2}      {1,2,3}
       {{1},{1}}  {{1},{1,2}}
       {{1},{2}}  {{1},{2,3}}
                  {{2},{1,3}}
                  {{3},{1,2}}
                  {{1},{1},{1}}
                  {{1},{1},{2}}
                  {{1},{2},{3}}
                  {{1},{{1},{1}}}
                  {{1},{{1},{2}}}
                  {{1},{{2},{3}}}
                  {{2},{{1},{1}}}
                  {{2},{{1},{3}}}
                  {{3},{{1},{2}}}

The generalization where the leaves are multisets is A330467.
The singleton-reduced case is A330628.
The unlabeled version is A330624.
The case with all atoms distinct is A005804.
The case with all atoms equal is A196545.
The case where all leaves are singletons is A330471.
Cf. A000669, A004111, A141268, A300660, A316652, A330469, A330475, A330626, A330668, A330675.

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];
srtrees[m_]:=Prepend[Join@@Table[Tuples[srtrees/@p],{p,Select[mps[m],Length[#1]>1&]}],m];
Table[Sum[Length[Select[srtrees[s],FreeQ[#,{_,x_Integer,x_Integer,_}]&]],{s,strnorm[n]}],{n,0,5}]

A330626 Number of non-isomorphic series/singleton-reduced rooted trees whose leaves are sets (not necessarily disjoint) with a total of n atoms.

Original entry on oeis.org

1, 1, 1, 3, 17, 100, 755

Offset: 0

Gus Wiseman, Dec 26 2019

A series/singleton-reduced rooted tree on a multiset m is either the multiset m itself or a sequence of series/singleton-reduced rooted trees, one on each part of a multiset partition of m that is neither minimal (all singletons) nor maximal (only one part).

			Non-isomorphic representatives of the a(1) = 1 through a(4) = 17 trees:
  {1}  {1,2}  {1,2,3}      {1,2,3,4}
              {{1},{1,2}}  {{1},{1,2,3}}
              {{1},{2,3}}  {{1,2},{1,2}}
                           {{1,2},{1,3}}
                           {{1},{2,3,4}}
                           {{1,2},{3,4}}
                           {{1},{1},{1,2}}
                           {{1},{1},{2,3}}
                           {{1},{2},{1,2}}
                           {{1},{2},{1,3}}
                           {{1},{2},{3,4}}
                           {{1},{{1},{1,2}}}
                           {{1},{{1},{2,3}}}
                           {{1},{{2},{1,2}}}
                           {{1},{{2},{1,3}}}
                           {{1},{{2},{3,4}}}
                           {{2},{{1},{1,3}}}

The non-singleton-reduced version is A330624.
The generalization where leaves are multisets is A330470.
A labeled version is A330628 (strongly normal).
The case with all atoms distinct is A004114.
The balanced version is A330668.
Cf. A000669, A004111, A005804, A007716, A141268, A330465, A330625, A330627, A330654, A330663, A330677.

A330628 Number of series/singleton-reduced rooted trees on strongly normal multisets of size n whose leaves are sets (not necessarily disjoint).

Original entry on oeis.org

1, 1, 1, 5, 42, 423, 5458, 80926

Offset: 0

Gus Wiseman, Dec 26 2019

A series/singleton-reduced rooted tree on a multiset m is either the multiset m itself or a sequence of series/singleton-reduced rooted trees, one on each part of a multiset partition of m that is neither minimal (all singletons) nor maximal (only one part).

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

			The a(4) = 42 trees:
  {{1}{1}{12}}    {{12}{12}}      {{1}{123}}      {1234}
  {{1}{{1}{12}}}  {{1}{2}{12}}    {{12}{13}}      {{1}{234}}
                  {{1}{{2}{12}}}  {{1}{1}{23}}    {{12}{34}}
                  {{2}{{1}{12}}}  {{1}{2}{13}}    {{13}{24}}
                                  {{1}{3}{12}}    {{14}{23}}
                                  {{1}{{1}{23}}}  {{2}{134}}
                                  {{1}{{2}{13}}}  {{3}{124}}
                                  {{1}{{3}{12}}}  {{4}{123}}
                                  {{2}{{1}{13}}}  {{1}{2}{34}}
                                  {{3}{{1}{12}}}  {{1}{3}{24}}
                                                  {{1}{4}{23}}
                                                  {{2}{3}{14}}
                                                  {{2}{4}{13}}
                                                  {{3}{4}{12}}
                                                  {{1}{{2}{34}}}
                                                  {{1}{{3}{24}}}
                                                  {{1}{{4}{23}}}
                                                  {{2}{{1}{34}}}
                                                  {{2}{{3}{14}}}
                                                  {{2}{{4}{13}}}
                                                  {{3}{{1}{24}}}
                                                  {{3}{{2}{14}}}
                                                  {{3}{{4}{12}}}
                                                  {{4}{{1}{23}}}
                                                  {{4}{{2}{13}}}
                                                  {{4}{{3}{12}}}

The generalization where leaves are multisets is A330471.
The non-singleton-reduced version is A330625.
The unlabeled version is A330626.
The case with all atoms distinct is A000311.
Strongly normal multiset partitions are A035310.
Cf. A000669, A004111, A004114, A005804, A196545, A281118, A330465, A330467, A330624, A330654, A330668.

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];
ssrtrees[m_]:=Prepend[Join@@Table[Tuples[ssrtrees/@p],{p,Select[mps[m],Length[m]>Length[#1]>1&]}],m];
Table[Sum[Length[Select[ssrtrees[s],FreeQ[#,{_,x_Integer,x_Integer,_}]&]],{s,strnorm[n]}],{n,0,5}]

A330677 Number of non-isomorphic balanced reduced multisystems of weight n and maximum depth whose leaves (which are multisets of atoms) are sets.

Original entry on oeis.org

1, 1, 1, 2, 11, 81, 859

Offset: 0

Gus Wiseman, Dec 30 2019

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. The weight of an atom is 1, while the weight of a multiset is the sum of weights of its elements.

			Non-isomorphic representatives of the a(0) = 1 through a(4) = 11 multisystems:
  {}  {1}  {1,2}  {{1},{1,2}}  {{{1}},{{1},{1,2}}}
                  {{1},{2,3}}  {{{1}},{{1},{2,3}}}
                               {{{1,2}},{{1},{1}}}
                               {{{1}},{{2},{1,2}}}
                               {{{1,2}},{{1},{2}}}
                               {{{1}},{{2},{1,3}}}
                               {{{1,2}},{{1},{3}}}
                               {{{1}},{{2},{3,4}}}
                               {{{1,2}},{{3},{4}}}
                               {{{2}},{{1},{1,3}}}
                               {{{2,3}},{{1},{1}}}

The version with all distinct atoms is A000111.
Non-isomorphic set multipartitions are A049311.
The (non-maximal) tree version is A330626.
Allowing leaves to be multisets gives A330663.
The case with prescribed degrees is A330664.
The version allowing all depths is A330668.
Cf. A000669, A001678, A004114, A005121, A007716, A141268, A283877, A306186, A330465, A330470, A330624.

cp's OEIS Frontend

A330474 Number of non-isomorphic balanced reduced multisystems of weight n.

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A330624 Number of non-isomorphic series-reduced rooted trees whose leaves are sets (not necessarily disjoint) with a total of n elements.

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A330625 Number of series-reduced rooted trees whose leaves are sets (not necessarily disjoint) with multiset union a strongly normal multiset of size n.

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A330626 Number of non-isomorphic series/singleton-reduced rooted trees whose leaves are sets (not necessarily disjoint) with a total of n atoms.

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A330628 Number of series/singleton-reduced rooted trees on strongly normal multisets of size n whose leaves are sets (not necessarily disjoint).

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A330677 Number of non-isomorphic balanced reduced multisystems of weight n and maximum depth whose leaves (which are multisets of atoms) are sets.

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