cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A320449 Number of antichains of sets whose multiset union is an integer partition of n.

Original entry on oeis.org

1, 1, 2, 4, 6, 9, 18, 24, 39, 58, 92, 131, 206
Offset: 0

Views

Author

Gus Wiseman, Oct 12 2018

Keywords

Examples

			The a(1) = 1 through a(7) = 24 antichains:
  {{1}}  {{2}}      {{3}}          {{4}}              {{5}}
         {{1},{1}}  {{1,2}}        {{1,3}}            {{1,4}}
                    {{1},{2}}      {{1},{3}}          {{2,3}}
                    {{1},{1},{1}}  {{2},{2}}          {{1},{4}}
                                   {{1},{1},{2}}      {{2},{3}}
                                   {{1},{1},{1},{1}}  {{1},{1},{3}}
                                                      {{1},{2},{2}}
                                                      {{1},{1},{1},{2}}
                                                      {{1},{1},{1},{1},{1}}
.
  {{6}}                      {{7}}
  {{1,5}}                    {{1,6}}
  {{2,4}}                    {{2,5}}
  {{1,2,3}}                  {{3,4}}
  {{1},{5}}                  {{1,2,4}}
  {{2},{4}}                  {{1},{6}}
  {{3},{3}}                  {{2},{5}}
  {{1},{2,3}}                {{3},{4}}
  {{2},{1,3}}                {{1},{2,4}}
  {{3},{1,2}}                {{2},{1,4}}
  {{1},{1},{4}}              {{4},{1,2}}
  {{1,2},{1,2}}              {{1},{1},{5}}
  {{1},{2},{3}}              {{1,2},{1,3}}
  {{2},{2},{2}}              {{1},{2},{4}}
  {{1},{1},{1},{3}}          {{1},{3},{3}}
  {{1},{1},{2},{2}}          {{2},{2},{3}}
  {{1},{1},{1},{1},{2}}      {{1},{1},{2,3}}
  {{1},{1},{1},{1},{1},{1}}  {{1},{1},{1},{4}}
                             {{1},{1},{2},{3}}
                             {{1},{2},{2},{2}}
                             {{1},{1},{1},{1},{3}}
                             {{1},{1},{1},{2},{2}}
                             {{1},{1},{1},{1},{1},{2}}
                             {{1},{1},{1},{1},{1},{1},{1}}

Crossrefs

Cf. A001970, A089259, A258466, A319719, A319721, A320328, A320353, A320355, A320356.

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]]]];
submultisetQ[M_,N_]:=Or[Length[M]==0,MatchQ[{Sort[List@@M],Sort[List@@N]},{{x_,Z___},{_,x_,W___}}/;submultisetQ[{Z},{W}]]];
antiQ[s_]:=Select[Tuples[s,2],And[UnsameQ@@#,submultisetQ@@#]&]=={};
Table[Length[Select[Join@@mps/@IntegerPartitions[n],And[And@@UnsameQ@@@#,antiQ[#]]&]],{n,10}]

A320353 Number of antichains of multisets whose multiset union is an integer partition of n.

Original entry on oeis.org

1, 1, 3, 5, 11, 17, 36, 56, 107, 175, 311, 505, 887
Offset: 0

Views

Author

Gus Wiseman, Oct 11 2018

Keywords

Examples

			The a(1) = 1 through a(5) = 17 antichains:
  {{1}}  {{2}}      {{3}}          {{4}}              {{5}}
         {{1,1}}    {{1,2}}        {{1,3}}            {{1,4}}
         {{1},{1}}  {{1,1,1}}      {{2,2}}            {{2,3}}
                    {{1},{2}}      {{1,1,2}}          {{1,1,3}}
                    {{1},{1},{1}}  {{1},{3}}          {{1,2,2}}
                                   {{2},{2}}          {{1},{4}}
                                   {{1,1,1,1}}        {{2},{3}}
                                   {{2},{1,1}}        {{1,1,1,2}}
                                   {{1,1},{1,1}}      {{1},{2,2}}
                                   {{1},{1},{2}}      {{3},{1,1}}
                                   {{1},{1},{1},{1}}  {{1,1,1,1,1}}
                                                      {{1,1},{1,2}}
                                                      {{1},{1},{3}}
                                                      {{1},{2},{2}}
                                                      {{2},{1,1,1}}
                                                      {{1},{1},{1},{2}}
                                                      {{1},{1},{1},{1},{1}}

Links

Crossrefs

Cf. A001970, A006126, A050342, A089259, A258466, A261005, A261049, A293994, A319719, A319721, A320328, A320355, A320356.

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]]]];
submultisetQ[M_,N_]:=Or[Length[M]==0,MatchQ[{Sort[List@@M],Sort[List@@N]},{{x_,Z___},{_,x_,W___}}/;submultisetQ[{Z},{W}]]];
antiQ[s_]:=Select[Tuples[s,2],And[UnsameQ@@#,submultisetQ@@#]&]=={};
Table[Length[Select[Join@@mps/@IntegerPartitions[n],antiQ]],{n,8}]

A320356 Number of strict connected antichains of multisets whose multiset union is an integer partition of n.

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 13, 22, 35, 62, 98, 171, 277
Offset: 0

Views

Author

Gus Wiseman, Oct 11 2018

Keywords

Examples

			The a(1) = 1 through a(6) = 13 clutters:
  {{1}}  {{2}}    {{3}}      {{4}}        {{5}}          {{6}}
         {{1,1}}  {{1,2}}    {{1,3}}      {{1,4}}        {{1,5}}
                  {{1,1,1}}  {{2,2}}      {{2,3}}        {{2,4}}
                             {{1,1,2}}    {{1,1,3}}      {{3,3}}
                             {{1,1,1,1}}  {{1,2,2}}      {{1,1,4}}
                                          {{1,1,1,2}}    {{1,2,3}}
                                          {{1,1,1,1,1}}  {{2,2,2}}
                                          {{1,1},{1,2}}  {{1,1,1,3}}
                                                         {{1,1,2,2}}
                                                         {{1,1,1,1,2}}
                                                         {{1,1},{1,3}}
                                                         {{1,1,1,1,1,1}}
                                                         {{1,2},{1,1,1}}

Links

Crossrefs

Cf. A001970, A007718, A048143, A056156, A258466, A261006, A293994, A318403, A319079, A319719, A319721, A320351, A320353, A320355.

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]]]];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Union[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
submultisetQ[M_,N_]:=Or[Length[M]==0,MatchQ[{Sort[List@@M],Sort[List@@N]},{{x_,Z___},{_,x_,W___}}/;submultisetQ[{Z},{W}]]];
antiQ[s_]:=Select[Tuples[s,2],And[UnsameQ@@#,submultisetQ@@#]&]=={};
Table[Length[Select[Join@@mps/@IntegerPartitions[n],And[UnsameQ@@#,Length[csm[#]]==1,antiQ[#]]&]],{n,8}]

A320351 Number of connected multiset partitions of integer partitions of n.

Original entry on oeis.org

1, 1, 3, 5, 11, 18, 38, 66, 130, 237, 449, 823, 1538
Offset: 0

Views

Author

Gus Wiseman, Oct 11 2018

Keywords

Examples

			The a(1) = 1 through a(5) = 18 multiset partitions:
  {{1}}  {{2}}      {{3}}          {{4}}              {{5}}
         {{1,1}}    {{1,2}}        {{1,3}}            {{1,4}}
         {{1},{1}}  {{1,1,1}}      {{2,2}}            {{2,3}}
                    {{1},{1,1}}    {{1,1,2}}          {{1,1,3}}
                    {{1},{1},{1}}  {{2},{2}}          {{1,2,2}}
                                   {{1,1,1,1}}        {{1,1,1,2}}
                                   {{1},{1,2}}        {{1},{1,3}}
                                   {{1},{1,1,1}}      {{2},{1,2}}
                                   {{1,1},{1,1}}      {{1,1,1,1,1}}
                                   {{1},{1},{1,1}}    {{1},{1,1,2}}
                                   {{1},{1},{1},{1}}  {{1,1},{1,2}}
                                                      {{1},{1,1,1,1}}
                                                      {{1,1},{1,1,1}}
                                                      {{1},{1},{1,2}}
                                                      {{1},{1},{1,1,1}}
                                                      {{1},{1,1},{1,1}}
                                                      {{1},{1},{1},{1,1}}
                                                      {{1},{1},{1},{1},{1}}

Crossrefs

Cf. A001970, A007718, A048143, A056156, A258466, A261006, A293994, A319719, A320328, A320330, A320331, A320355, A320356.

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]]]];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Union[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Table[Length[Select[Join@@mps/@IntegerPartitions[n],Length[csm[#]]==1&]],{n,8}]

A318403 Number of strict connected antichains of sets whose multiset union is an integer partition of n.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 6, 8, 12, 13, 22, 31
Offset: 0

Views

Author

Gus Wiseman, Oct 12 2018

Keywords

Examples

			The a(1) = 1 through a(10) = 13 clutters:
  {{1}}  {{2}}  {{3}}    {{4}}    {{5}}    {{6}}      {{7}}
                {{1,2}}  {{1,3}}  {{1,4}}  {{1,5}}    {{1,6}}
                                  {{2,3}}  {{2,4}}    {{2,5}}
                                           {{1,2,3}}  {{3,4}}
                                                      {{1,2,4}}
                                                      {{1,2},{1,3}}
.
  {{8}}          {{9}}          {{10}}
  {{1,7}}        {{1,8}}        {{1,9}}
  {{2,6}}        {{2,7}}        {{2,8}}
  {{3,5}}        {{3,6}}        {{3,7}}
  {{1,2,5}}      {{4,5}}        {{4,6}}
  {{1,3,4}}      {{1,2,6}}      {{1,2,7}}
  {{1,2},{1,4}}  {{1,3,5}}      {{1,3,6}}
  {{1,2},{2,3}}  {{2,3,4}}      {{1,4,5}}
                 {{1,2},{1,5}}  {{2,3,5}}
                 {{1,2},{2,4}}  {{1,2,3,4}}
                 {{1,3},{1,4}}  {{1,2},{1,6}}
                 {{1,3},{2,3}}  {{1,2},{2,5}}
                                {{1,3},{1,5}}

Crossrefs

Cf. A001970, A007718, A048143, A050342, A056156, A089259, A261049, A293994, A319719, A320351, A320353, A320355, A320356.

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]]]];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Union[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
submultisetQ[M_,N_]:=Or[Length[M]==0,MatchQ[{Sort[List@@M],Sort[List@@N]},{{x_,Z___},{_,x_,W___}}/;submultisetQ[{Z},{W}]]];
antiQ[s_]:=Select[Tuples[s,2],And[UnsameQ@@#,submultisetQ@@#]&]=={};
Table[Length[Select[Join@@mps/@IntegerPartitions[n],And[UnsameQ@@#,And@@UnsameQ@@@#,Length[csm[#]]==1,antiQ[#]]&]],{n,8}]

A319079 Number of connected antichains of sets whose multiset union is an integer partition of n.

Original entry on oeis.org

1, 1, 2, 3, 4, 4, 8, 7, 12, 15, 19, 26, 43
Offset: 0

Views

Author

Gus Wiseman, Oct 12 2018

Keywords

Examples

			The a(10) = 19 clutters:
  {{10}}
  {{1,9}}
  {{2,8}}
  {{3,7}}
  {{4,6}}
  {{1,2,7}}
  {{1,3,6}}
  {{1,4,5}}
  {{2,3,5}}
  {{1,2,3,4}}
  {{5},{5}}
  {{1,2},{1,6}}
  {{1,2},{2,5}}
  {{1,3},{1,5}}
  {{1,4},{1,4}}
  {{2,3},{2,3}}
  {{1,2},{1,2},{1,3}}
  {{2},{2},{2},{2},{2}}
  {{1},{1},{1},{1},{1},{1},{1},{1},{1},{1}}

Crossrefs

Cf. A001970, A007718, A048143, A056156, A089259, A319719, A320351, A320353, A320355, A320356.

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]]]];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Union[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
submultisetQ[M_,N_]:=Or[Length[M]==0,MatchQ[{Sort[List@@M],Sort[List@@N]},{{x_,Z___},{_,x_,W___}}/;submultisetQ[{Z},{W}]]];
antiQ[s_]:=Select[Tuples[s,2],And[UnsameQ@@#,submultisetQ@@#]&]=={};
Table[Length[Select[Join@@mps/@IntegerPartitions[n],And[And@@UnsameQ@@@#,Length[csm[#]]==1,antiQ[#]]&]],{n,10}]

A319255 Number of strict antichains of multisets whose multiset union is an integer partition of n.

Original entry on oeis.org

1, 1, 2, 4, 7, 13, 23, 41, 70, 123, 208, 355, 597
Offset: 0

Views

Author

Gus Wiseman, Oct 12 2018

Keywords

Examples

			The a(1) = 1 through a(6) = 23 antichains:
  {{1}}  {{2}}    {{3}}      {{4}}        {{5}}          {{6}}
         {{1,1}}  {{1,2}}    {{1,3}}      {{1,4}}        {{1,5}}
                  {{1,1,1}}  {{2,2}}      {{2,3}}        {{2,4}}
                  {{1},{2}}  {{1,1,2}}    {{1,1,3}}      {{3,3}}
                             {{1},{3}}    {{1,2,2}}      {{1,1,4}}
                             {{1,1,1,1}}  {{1},{4}}      {{1,2,3}}
                             {{2},{1,1}}  {{2},{3}}      {{1},{5}}
                                          {{1,1,1,2}}    {{2,2,2}}
                                          {{1},{2,2}}    {{2},{4}}
                                          {{3},{1,1}}    {{1,1,1,3}}
                                          {{1,1,1,1,1}}  {{1,1,2,2}}
                                          {{1,1},{1,2}}  {{1},{2,3}}
                                          {{2},{1,1,1}}  {{2},{1,3}}
                                                         {{3},{1,2}}
                                                         {{4},{1,1}}
                                                         {{1,1,1,1,2}}
                                                         {{1,1},{1,3}}
                                                         {{1,1},{2,2}}
                                                         {{1},{2},{3}}
                                                         {{3},{1,1,1}}
                                                         {{1,1,1,1,1,1}}
                                                         {{1,2},{1,1,1}}
                                                         {{2},{1,1,1,1}}

Links

Crossrefs

Cf. A001970, A258466, A261049, A319719, A319721, A320353, A320355, A320356.

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]]]];
submultisetQ[M_,N_]:=Or[Length[M]==0,MatchQ[{Sort[List@@M],Sort[List@@N]},{{x_,Z___},{_,x_,W___}}/;submultisetQ[{Z},{W}]]];
antiQ[s_]:=Select[Tuples[s,2],And[UnsameQ@@#,submultisetQ@@#]&]=={};
Table[Length[Select[Join@@mps/@IntegerPartitions[n],And[UnsameQ@@#,antiQ[#]]&]],{n,10}]
Showing 1-7 of 7 results.

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