A326293 -id:A326293 - cp's OEIS frontend

A326294 Number of connected simple graphs on a subset of {1..n} with no crossing or nesting edges.

1, 1, 2, 8, 35, 147, 600, 2418

Offset: 0

Gus Wiseman, Jun 29 2019

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.

			The a(4) = 35 edge-sets:
  {}  {12}  {12,13}  {12,13,14}  {12,13,14,34}
      {13}  {12,14}  {12,13,23}  {12,13,23,34}
      {14}  {12,23}  {12,13,34}  {12,14,24,34}
      {23}  {12,24}  {12,14,24}  {12,23,24,34}
      {24}  {13,14}  {12,14,34}
      {34}  {13,23}  {12,23,24}
            {13,34}  {12,23,34}
            {14,24}  {12,24,34}
            {14,34}  {13,14,34}
            {23,24}  {13,23,34}
            {23,34}  {14,24,34}
            {24,34}  {23,24,34}

Eric Marberg, Crossings and nestings in colored set partitions, arXiv preprint arXiv:1203.5738 [math.CO], 2012.

The inverse binomial transform is the covering case A326339.
Covering graphs with no crossing or nesting edges are A326329.
Connected simple graphs are A001349.
Graphs without crossing or nesting edges are A326244.
Cf. A006125, A054726, A117662, A136653.
Cf. A324169, A326210, A326293, A326340.

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[#]]<=1&&!MatchQ[#,{_,{x_,y_},_,{z_,t_},_}/;x

Conjecture: a(n) = A052161(n - 2) + 1.

A326341 Number of minimal topologically connected chord graphs covering {1..n}.

Original entry on oeis.org

1, 0, 1, 0, 1, 5, 22, 119

Offset: 0

Gus Wiseman, Jun 29 2019

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. A graph is topologically connected if the graph whose vertices are the edges and whose edges are crossing pairs of edges is connected.

			The a(4) = 1 through a(6) = 22 edge-sets:
  {13,24}  {13,14,25}  {13,25,46}
           {13,24,25}  {14,25,36}
           {13,24,35}  {14,26,35}
           {14,24,35}  {15,24,36}
           {14,25,35}  {13,14,15,26}
                       {13,14,25,26}
                       {13,15,24,26}
                       {13,15,26,46}
                       {13,24,25,26}
                       {13,24,25,36}
                       {13,24,26,35}
                       {13,24,35,36}
                       {13,24,35,46}
                       {14,15,26,36}
                       {14,24,35,36}
                       {14,24,35,46}
                       {14,25,35,46}
                       {15,24,35,46}
                       {15,25,35,46}
                       {15,25,36,46}
                       {15,26,35,46}
                       {15,26,36,46}

The non-minimal case is A324327.
Minimal covers are A053530.
Topologically connected graphs are A324327 (covering) or A324328 (all).
Cf. A000108, A006125, A007297, A054726, A136653, A324169, A326210, A326293.

croXQ[stn_]:=MatchQ[stn,{_,{_,x_,_,y_,_},_,{_,z_,_,t_,_},_}/;x0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
crosscmpts[stn_]:=csm[Union[Subsets[stn,{1}],Select[Subsets[stn,{2}],croXQ]]];
Table[Length[fasmin[Select[Subsets[Subsets[Range[n],{2}]],And[Union@@#==Range[n],Length[crosscmpts[#]]<=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

Gus Wiseman, Jun 30 2019

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.

			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}

Gus Wiseman, The a(6) = 95 non-nesting, topologically connected, covering simple graphs.

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.

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}]

A326350 Number of non-nesting connected simple graphs with vertices {1..n}.

Original entry on oeis.org

1, 0, 1, 4, 23, 157, 1182

Offset: 0

Gus Wiseman, Jun 30 2019

Two edges {a,b}, {c,d} are nesting if a < c < d < b or c < a < b < d.

Gus Wiseman, The a(4) = 23 non-nesting connected simple graphs.
Gus Wiseman, The a(5) = 157 non-nesting connected simple graphs.

The inverse binomial transform is the non-covering case A326351.
Connected simple graphs are A001349.
Connected simple graphs with no crossing or nesting edges are A326294.
Simple graphs without crossing or nesting edges are A326244.
Cf. A006125, A054726, A117662, A136653.
Cf. A324169, A326210, A326293, A326329, A326340.

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[csm[#]]<=1&&!MatchQ[#,{_,{x_,y_},_,{z_,t_},_}/;x

A326351 Number of non-nesting connected simple graphs on a subset of {1..n}.

Original entry on oeis.org

1, 1, 2, 8, 46, 323, 2565

Offset: 0

Gus Wiseman, Jun 30 2019

Two edges {a,b}, {c,d} are nesting if a < c < d < b or c < a < b < d.

The binomial transform is the covering case A326350.
Connected simple graphs are A001349.
Connected simple graphs with no crossing or nesting edges are A326294.
Simple graphs without crossing or nesting edges are A326244.
Cf. A006125, A054726, A117662, A136653.
Cf. A324169, A326210, A326293, A326329, A326340.

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[#]]<=1&&!MatchQ[#,{_,{x_,y_},_,{z_,t_},_}/;x

cp's OEIS Frontend

A326294 Number of connected simple graphs on a subset of {1..n} with no crossing or nesting edges.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A326341 Number of minimal topologically connected chord graphs covering {1..n}.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A326350 Number of non-nesting connected simple graphs with vertices {1..n}.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A326351 Number of non-nesting connected simple graphs on a subset of {1..n}.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica