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.

A326293 Number of non-nesting, topologically connected simple graphs with vertices {1..n}.

Original entry on oeis.org

1, 1, 2, 4, 8, 27, 192, 1750
Offset: 0

Views

Author

Gus Wiseman, Jun 29 2019

Keywords

Comments

Two edges {a,b}, {c,d} are crossing if a < c < b < d or c < a < d < b, and nesting if a < c < d < b or c < a < b < d. A graph with positive integer vertices is topologically connected if the graph whose vertices are the edges and whose edges are crossing pairs of edges is connected.

Links

Crossrefs

The inverse binomial transform is the covering case A326349.
Topologically connected simple graphs are A324328.
Non-crossing simple graphs are A054726.
Topologically connected set partitions are A099947.
Cf. A000108, A000699, A006125, A007297, A117662, A136653.
Cf. A324173, A324323, A324327, A326244, A326330, A326341.

Programs

  • Mathematica
    croXQ[eds_]:=MatchQ[eds,{_,{x_,y_},_,{z_,t_},_}/;x_,{x_,y_},_,{z_,t_},_}/;x0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],!nesXQ[#]&&Length[csm[Union[Subsets[#,{1}],Select[Subsets[#,{2}],croXQ]]]]<=1&]],{n,0,5}]

A326337 Number of simple graphs covering the vertices {1..n} whose weakly nesting edges are connected.

Original entry on oeis.org

1, 0, 1, 3, 29, 595, 23437
Offset: 0

Views

Author

Gus Wiseman, Jun 28 2019

Keywords

Comments

Two edges {a,b}, {c,d} are weakly nesting if a <= c < d <= b or c <= a < b <= d. A graph has its weakly nesting edges connected if the graph whose vertices are the edges and whose edges are weakly nesting pairs of edges is connected.

Links

Crossrefs

The binomial transform is the non-covering case A326338.
The non-weak case is A326331.
Simple graphs whose nesting edges are connected are A326330.
Cf. A006125, A007297, A054726, A099947, A136653.
Cf. A324169, A324327, A326293, A326294, A326329, A326335, A326336.

Programs

  • Mathematica
    wknXQ[stn_]:=MatchQ[stn,{_,{_,x_,y_,_},_,{_,z_,t_,_},_}/;(x<=z&&y>=t)||(x>=z&&y<=t)];
wknestcmpts[stn_]:=csm[Union[List/@stn,Select[Subsets[stn,{2}],wknXQ]]];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],Union@@#==Range[n]&&Length[wknestcmpts[#]]<=1&]],{n,0,5}]

A326331 Number of simple graphs covering the vertices {1..n} whose nesting edges are connected.

Original entry on oeis.org

1, 0, 1, 0, 1, 14, 539
Offset: 0

Views

Author

Gus Wiseman, Jun 27 2019

Keywords

Comments

Covering means there are no isolated vertices. Two edges {a,b}, {c,d} are nesting if a < c < d < b or c < a < b < d. A graph has its nesting edges connected if the graph whose vertices are the edges and whose edges are nesting pairs of edges is connected.

Links

Crossrefs

The non-covering case is the binomial transform A326330.
Covering graphs whose crossing edges are connected are A324327.
Cf. A006125, A007297, A054726, A099947, A117662, A136653, A324328.
Cf. A326210, A326293, A326335, A326336, A326337, A326338, A326339.

Programs

  • Mathematica
    nesXQ[stn_]:=MatchQ[stn,{_,{x_,y_},_,{z_,t_},_}/;x0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],Union@@#==Range[n]&&Length[nestcmpts[#]]<=1&]],{n,0,5}]

A326335 Number of set partitions of {1..n} whose nesting blocks are connected.

Original entry on oeis.org

1, 1, 1, 1, 2, 6, 21, 86, 394, 1974, 10696
Offset: 0

Views

Author

Gus Wiseman, Jun 27 2019

Keywords

Comments

Two blocks are nesting if they are of the form {...x,y...}, {...z,t...} where x < z < t < y or z < x < y < t. A set partition has its nesting blocks connected if the graph whose vertices are the blocks and whose edges are nesting pairs of blocks is connected.

Examples

			The a(0) = 1 through a(6) = 21 set partitions:
  {}  {1}  {12}  {123}  {1234}    {12345}    {123456}
                        {14}{23}  {125}{34}  {1236}{45}
                                  {134}{25}  {1245}{36}
                                  {14}{235}  {125}{346}
                                  {145}{23}  {1256}{34}
                                  {15}{234}  {126}{345}
                                             {134}{256}
                                             {1345}{26}
                                             {1346}{25}
                                             {136}{245}
                                             {14}{2356}
                                             {145}{236}
                                             {1456}{23}
                                             {146}{235}
                                             {15}{2346}
                                             {156}{234}
                                             {16}{2345}
                                             {15}{26}{34}
                                             {16}{23}{45}
                                             {16}{24}{35}
                                             {16}{25}{34}

Crossrefs

Simple graphs whose nesting blocks are connected are A326330.
Set partitions whose crossing blocks are connected are A099947.
Set partitions whose capturing blocks are connected are A326336.
Cf. A000110, A001519, A016098, A122880, A324173, A326243, A326248, A326293, A326331, A326337.

Programs

  • Mathematica
    nesXQ[stn_]:=MatchQ[stn,{_,{_,x_,y_,_},_,{_,z_,t_,_},_}/;x0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
nestcmpts[stn_]:=csm[Union[List/@stn,Select[Subsets[stn,{2}],nesXQ]]];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[sps[Range[n]],Length[nestcmpts[#]]<=1&]],{n,0,5}]

A326336 Number of set partitions of {1..n} whose capturing blocks are connected.

Original entry on oeis.org

1, 1, 1, 1, 2, 7, 24, 100, 458, 2279, 12270
Offset: 0

Views

Author

Gus Wiseman, Jun 28 2019

Keywords

Comments

Two blocks are capturing if they are of the form {...x...y...}, {...z...t...} where x < z < t < y or z < x < y < t. A set partition has its capturing blocks connected if the graph whose vertices are the blocks and whose edges are capturing pairs of blocks is connected.

Examples

			The a(0) = 1 through a(6) = 24 set partitions:
  {}  {1}  {12}  {123}  {1234}    {12345}    {123456}
                        {14}{23}  {125}{34}  {1236}{45}
                                  {134}{25}  {1245}{36}
                                  {135}{24}  {1246}{35}
                                  {14}{235}  {125}{346}
                                  {145}{23}  {1256}{34}
                                  {15}{234}  {126}{345}
                                             {134}{256}
                                             {1345}{26}
                                             {1346}{25}
                                             {135}{246}
                                             {1356}{24}
                                             {136}{245}
                                             {14}{2356}
                                             {145}{236}
                                             {1456}{23}
                                             {146}{235}
                                             {15}{2346}
                                             {156}{234}
                                             {16}{2345}
                                             {15}{26}{34}
                                             {16}{23}{45}
                                             {16}{24}{35}
                                             {16}{25}{34}

Crossrefs

Simple graphs whose capturing blocks are connected are A326330.
Set partitions whose crossing blocks are connected are A099947.
Set partitions whose nesting blocks are connected are A326335.
Cf. A000110, A001519, A016098, A122880, A324173, A326243, A326248, A326293, A326331, A326337.

Programs

  • Mathematica
    capXQ[stn_]:=MatchQ[stn,{_,{_,x_,_,y_,_},_,{_,z_,_,t_,_},_}/;x0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
captcmpts[stn_]:=csm[Union[List/@stn,Select[Subsets[stn,{2}],capXQ]]];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[sps[Range[n]],Length[captcmpts[#]]<=1&]],{n,0,6}]

A326338 Number of simple graphs with vertices {1..n} whose weakly nesting edges are connected.

Original entry on oeis.org

1, 1, 2, 7, 48, 781, 27518
Offset: 0

Views

Author

Gus Wiseman, Jun 29 2019

Keywords

Comments

Two edges {a,b}, {c,d} are weakly nesting if a <= c < d <= b or c <= a < b <= d. A graph has its weakly nesting edges connected if the graph whose vertices are the edges and whose edges are weakly nesting pairs of edges is connected.

Links

Crossrefs

The inverse binomial transform is the covering case A326337.
The non-weak case is A326330.
Cf. A006125, A099947, A136653, A324328, A326289, A326293, A326331, A326335, A326336, A326341.

Programs

  • Mathematica
    wknXQ[eds_]:=MatchQ[eds,{_,{x_,y_},_,{z_,t_},_}/;(x<=z&&y>=t)||(x>=z&&y<=t)];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],Length[csm[Union[List/@#,Select[Subsets[#,{2}],wknXQ]]]]<=1&]],{n,0,5}]

A326349 Number of non-nesting, topologically connected simple graphs covering {1..n}.

Original entry on oeis.org

1, 0, 1, 0, 1, 11, 95, 797
Offset: 0

Views

Author

Gus Wiseman, Jun 30 2019

Keywords

Comments

Covering means there are no isolated vertices. Two edges {a,b}, {c,d} are crossing if a < c < b < d or c < a < d < b, and nesting if a < c < d < b or c < a < b < d. A graph with positive integer vertices is topologically connected if the graph whose vertices are the edges and whose edges are crossing pairs of edges is connected.

Examples

			The a(5) = 11 edge-sets:
  {13,14,25}
  {13,24,25}
  {13,24,35}
  {14,24,35}
  {14,25,35}
  {13,14,24,25}
  {13,14,24,35}
  {13,14,25,35}
  {13,24,25,35}
  {14,24,25,35}
  {13,14,24,25,35}

Links

Crossrefs

The binomial transform is the non-covering case A326293.
Topologically connected, covering simple graphs are A324327.
Non-crossing, covering simple graphs are A324169.
Cf. A000108, A000699, A006125, A054726, A099947, A117662.
Cf. A324323, A324328, A326329, A326330, A326339, A326341.

Programs

  • Mathematica
    croXQ[eds_]:=MatchQ[eds,{_,{x_,y_},_,{z_,t_},_}/;x_,{x_,y_},_,{z_,t_},_}/;x0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],Union@@#==Range[n]&&!nesXQ[#]&&Length[csm[Union[Subsets[#,{1}],Select[Subsets[#,{2}],croXQ]]]]<=1&]],{n,0,5}]
Showing 1-7 of 7 results.

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