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 1-3 of 3 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}]

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 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.