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 1-5 of 5 results.

A326330 Number of simple graphs with vertices {1..n} whose nesting edges are connected.

Original entry on oeis.org

1, 1, 2, 4, 8, 30, 654
Offset: 0

Views

Author

Gus Wiseman, Jun 27 2019

Keywords

Comments

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 covering case is the inverse binomial transform A326331.
Graphs whose crossing edges are connected are A324328.
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}]],Length[nestcmpts[#]]<=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}]

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}]
Showing 1-5 of 5 results.

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