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.

Previous Showing 11-17 of 17 results.

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

Original entry on oeis.org

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

Examples

			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}

Links

Crossrefs

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.

Programs

  • Mathematica
    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

Formula

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

A326247 Number of labeled n-vertex 2-edge multigraphs that are neither crossing nor nesting.

Original entry on oeis.org

0, 0, 1, 9, 32, 80, 165, 301, 504, 792, 1185, 1705, 2376, 3224, 4277, 5565, 7120, 8976, 11169, 13737, 16720, 20160, 24101, 28589, 33672, 39400, 45825, 53001, 60984, 69832, 79605, 90365, 102176, 115104, 129217, 144585, 161280, 179376, 198949, 220077, 242840
Offset: 0

Views

Author

Gus Wiseman, Jun 20 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.

Examples

			The a(3) = 9 pairs of edges:
  {12,12}
  {12,13}
  {12,23}
  {13,12}
  {13,13}
  {13,23}
  {23,12}
  {23,13}
  {23,23}

Links

Crossrefs

The case for simple graphs (rather than multigraphs) is A095661.
Simple graphs that are neither crossing nor nesting are A326244.
The case for set partitions is A001519.
Non-crossing and non-nesting simple graphs are (both) A054726.
Cf. A000108, A002061, A006125, A117662, A324170, A326250, A326256.

Programs

  • Mathematica
    croXQ[stn_]:=MatchQ[stn,{_,{x_,y_},_,{z_,t_},_}/;x_,{x_,y_},_,{z_,t_},_}/;x

Formula

Conjectures from Colin Barker, Jun 21 2019: (Start)
G.f.: x^2*(1 + 4*x - 3*x^2) / (1 - x)^5.
a(n) = (n*(12 - 19*n + 6*n^2 + n^3)) / 12.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>4.
(End)

A326289 a(0) = 0, a(n) = 2^binomial(n,2) - 2^(n - 1).

Original entry on oeis.org

0, 0, 0, 4, 56, 1008, 32736, 2097088, 268435328, 68719476480, 35184372088320, 36028797018962944, 73786976294838204416, 302231454903657293672448, 2475880078570760549798240256, 40564819207303340847894502555648, 1329227995784915872903807060280311808
Offset: 0

Views

Author

Gus Wiseman, Jun 23 2019

Keywords

Comments

Number of simple graphs with vertices {1..n} containing two edges {a,b}, {c,d} that are weakly crossing, meaning a <= c < b <= d or c <= a < d <= b.

Examples

			The a(4) = 56 weakly crossing edge-sets:
  {12,13}  {12,13,14}  {12,13,14,23}  {12,13,14,23,24}  {12,13,14,23,24,34}
  {12,14}  {12,13,23}  {12,13,14,24}  {12,13,14,23,34}
  {12,23}  {12,13,24}  {12,13,14,34}  {12,13,14,24,34}
  {12,24}  {12,13,34}  {12,13,23,24}  {12,13,23,24,34}
  {12,34}  {12,14,23}  {12,13,23,34}  {12,14,23,24,34}
  {13,14}  {12,14,24}  {12,13,24,34}  {13,14,23,24,34}
  {13,23}  {12,14,34}  {12,14,23,24}
  {13,24}  {12,23,24}  {12,14,23,34}
  {13,34}  {12,23,34}  {12,14,24,34}
  {14,24}  {12,24,34}  {12,23,24,34}
  {14,34}  {13,14,23}  {13,14,23,24}
  {23,24}  {13,14,24}  {13,14,23,34}
  {23,34}  {13,14,34}  {13,14,24,34}
  {24,34}  {13,23,24}  {13,23,24,34}
           {13,23,34}  {14,23,24,34}
           {13,24,34}
           {14,23,24}
           {14,23,34}
           {14,24,34}
           {23,24,34}

Crossrefs

Cf. A000088, A000108, A002662, A006125, A016098, A054726, A324170.
Cf. A326210, A326244, A326248, A326250, A326257, A326279, A326290.

Programs

  • Mathematica
    Table[If[n==0,0,2^Binomial[n,2]-2^(n-1)],{n,0,5}]

A326290 Number of non-crossing n-vertex graphs with loops.

Original entry on oeis.org

1, 2, 8, 64, 768, 11264, 184320, 3227648, 59179008, 1121714176, 21803040768, 432218832896, 8705009516544, 177618573852672, 3663840373899264, 76277945940836352, 1600706475536154624, 33823752545680490496, 719051629204296695808, 15368152475218787434496
Offset: 0

Views

Author

Gus Wiseman, Sep 12 2019

Keywords

Comments

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

Examples

			The a(0) = 1 through a(2) = 8 non-crossing edge sets with loops:
  {}  {}    {}
      {11}  {11}
            {12}
            {22}
            {11,12}
            {11,22}
            {12,22}
            {11,12,22}

Links

Crossrefs

Crossing and nesting simple graphs are (both) A326210, while non-crossing, non-nesting simple graphs are A326244.
Cf. A000108, A002061, A002662, A054726, A324172, A326252.

Programs

  • Mathematica
    croXQ[stn_]:=MatchQ[stn,{_,{x_,y_},_,{z_,t_},_}/;x
  • PARI
    seq(n)=Vec(1+3*x-4*x^2 -x*sqrt(1-24*x+16*x^2 + O(x^n))) \\ Andrew Howroyd, Sep 14 2019

Formula

From Andrew Howroyd, Sep 14 2019: (Start)
a(n) = 2^n * A054726(n).
G.f.: 1 + 3*x - 4*x^2 - x*sqrt(1 - 24*x + 16*x^2). (End)

Extensions

Terms a(6) and beyond from Andrew Howroyd, Sep 14 2019

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

Views

Author

Gus Wiseman, Jun 30 2019

Keywords

Comments

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

Links

Crossrefs

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.

Programs

  • Mathematica
    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

Views

Author

Gus Wiseman, Jun 30 2019

Keywords

Comments

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

Crossrefs

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.

Programs

  • Mathematica
    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

A326278 Number of n-vertex, 2-edge multigraphs that are not nesting. Number of n-vertex, 2-edge multigraphs that are not crossing.

Original entry on oeis.org

0, 0, 1, 9, 34, 90, 195, 371, 644, 1044, 1605, 2365, 3366, 4654, 6279, 8295, 10760, 13736, 17289, 21489, 26410, 32130, 38731, 46299, 54924, 64700, 75725, 88101, 101934, 117334, 134415, 153295, 174096, 196944, 221969, 249305, 279090, 311466, 346579, 384579
Offset: 0

Views

Author

Gus Wiseman, Jun 23 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.

Examples

			The a(3) = 9 non-crossing multigraphs:
  {12,12}
  {12,13}
  {12,23}
  {13,12}
  {13,13}
  {13,23}
  {23,12}
  {23,13}
  {23,23}

Crossrefs

A326247(n) <= a(n) <= A000537(n).
The case for 2-edge simple graphs (rather than multigraphs) is A117662.
Cf. A000108, A001519, A006125, A016098, A054726, A095661.
Cf. A326210, A326243, A326244, A326248, A326250.

Programs

  • Mathematica
    croXQ[stn_]:=MatchQ[stn,{_,{x_,y_},_,{z_,t_},_}/;x

Formula

Conjectures from Colin Barker, Jun 25 2019: (Start)
G.f.: x^2*(1 + 4*x - x^2) / (1 - x)^5.
a(n) = (n*(3 - 4*n + n^3)) / 6 .
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>4.
(End)
Previous Showing 11-17 of 17 results.

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