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 1-10 of 12 results. Next

A326225 Number of Hamiltonian unlabeled n-vertex digraphs (without loops).

Original entry on oeis.org

0, 1, 1, 4, 61, 3725, 844141, 626078904
Offset: 0

Views

Author

Gus Wiseman, Jun 15 2019

Keywords

Comments

A digraph is Hamiltonian if it contains a directed cycle passing through every vertex exactly once.

Examples

			Non-isomorphic representatives of the a(3) = 4 digraph edge-sets:
  {12,23,31}
  {12,13,21,32}
  {12,13,21,23,31}
  {12,13,21,23,31,32}

Links

Crossrefs

The labeled case is A326219.
The case with loops is A326226.
The undirected case is A003216.
Non-Hamiltonian unlabeled digraphs (without loops) are A326222.
Cf. A000595, A002416, A003087, A003216, A053763, A246446, A326204, A326218, A326223.

Extensions

a(5)-a(7) from Sean A. Irvine, Jun 16 2019

A326208 Number of Hamiltonian labeled simple graphs with n vertices.

Original entry on oeis.org

0, 1, 0, 1, 10, 218, 10078, 896756, 151676112, 47754337568, 28229412456056, 31665593711174080
Offset: 0

Views

Author

Gus Wiseman, Jun 15 2019

Keywords

Comments

A graph is Hamiltonian if it contains a cycle passing through every vertex exactly once.

Links

Crossrefs

The unlabeled version is A003216.
The directed version is A326204 (with loops) or A326219 (without loops).
Simple graphs not containing a Hamiltonian cycle are A326207.
Simple graphs containing a Hamiltonian path are A326206.
Cf. A003216, A006125, A057864, A283420.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Subsets[Range[n],{2}]],FindHamiltonianCycle[Graph[Range[n],#]]!={}&]],{n,0,4}] (* Mathematica 8.0+ *)

Formula

A006125(n) = a(n) + A326207(n).

Extensions

a(7)-a(11) added using tinygraph by Falk Hüffner, Jun 21 2019

A326226 Number of unlabeled n-vertex Hamiltonian digraphs (with loops).

Original entry on oeis.org

0, 2, 3, 24, 858
Offset: 0

Views

Author

Gus Wiseman, Jun 14 2019

Keywords

Comments

A digraph is Hamiltonian if it contains a directed cycle passing through every vertex exactly once.

Examples

			Non-isomorphic representatives of the a(2) = 3 digraph edge-sets:
  {12,21}
  {11,12,21}
  {11,12,21,22}

Links

Crossrefs

The labeled case is A326204.
The case without loops is A326225.
The undirected case is A003216 (without loops) or A326215 (with loops).
Unlabeled non-Hamiltonian digraphs are A326223.
Unlabeled digraphs with a Hamiltonian path are A326221.
Cf. A000595, A002416, A003087, A057864, A246446, A326208.

Programs

  • Mathematica
    dinorm[m_]:=If[m=={},{},If[Union@@m!=Range[Max@@Flatten[m]],dinorm[m/. Apply[Rule,Table[{(Union@@m)[[i]],i},{i,Length[Union@@m]}],{1}]],First[Sort[dinorm[m,1]]]]];
dinorm[m_,aft_]:=If[Length[Union@@m]<=aft,{m},With[{mx=Table[Count[m,i,{2}],{i,Select[Union@@m,#1>=aft&]}]},Union@@(dinorm[#1,aft+1]&)/@Union[Table[Map[Sort,m/. {par+aft-1->aft,aft->par+aft-1},{0}],{par,First/@Position[mx,Max[mx]]}]]]];
Table[Length[Select[Union[dinorm/@Subsets[Tuples[Range[n],2]]],FindHamiltonianCycle[Graph[Range[n],DirectedEdge@@@#]]!={}&]],{n,0,4}] (* Mathematica 8.0+. Warning: Using HamiltonianGraphQ instead of FindHamiltonianCycle returns a(4) = 867 which is incorrect *)

A326220 Number of non-Hamiltonian labeled n-vertex digraphs (with loops).

Original entry on oeis.org

1, 0, 12, 392, 46432, 20023232, 30595305216
Offset: 0

Views

Author

Gus Wiseman, Jun 15 2019

Keywords

Comments

A digraph is Hamiltonian if it contains a directed cycle passing through every vertex exactly once.

Examples

			The a(2) = 12 digraph edge-sets:
  {}  {11}  {11,12}  {11,12,22}
      {12}  {11,21}  {11,21,22}
      {21}  {11,22}
      {22}  {12,22}
            {21,22}

Links

Crossrefs

The unlabeled case is A326223.
The undirected case is A326239 (with loops) or A326207 (without loops).
The case without loops is A326218.
Digraphs (with loops) containing a Hamiltonian cycle are A326204.
Digraphs (with loops) not containing a Hamiltonian path are A326213.
Cf. A000595, A002416, A003024, A003216, A246446, A326208, A326219, A326222, A326224.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Tuples[Range[n],2]],FindHamiltonianCycle[Graph[Range[n],DirectedEdge@@@#]]=={}&]],{n,4}] (* Mathematica 8.0+. Warning: Using HamiltonianGraphQ instead of FindHamiltonianCycle returns a(4) = 46336 which is incorrect *)

Extensions

a(5)-a(6) from Bert Dobbelaere, Jun 11 2024

A326223 Number of non-Hamiltonian unlabeled n-vertex digraphs (with loops).

Original entry on oeis.org

1, 0, 7, 80, 2186
Offset: 0

Views

Author

Gus Wiseman, Jun 15 2019

Keywords

Comments

A digraph is Hamiltonian if it contains a directed cycle passing through every vertex exactly once.

Examples

			Non-isomorphic representatives of the a(2) = 7 digraph edge-sets:
  {}
  {11}
  {12}
  {11,12}
  {11,21}
  {11,22}
  {11,12,22}

Links

Crossrefs

The labeled case is A326220.
The case without loops is A326222.
The undirected case is A246446 (without loops) or A326239 (with loops).
Hamiltonian unlabeled digraphs are A326226.
Unlabeled digraphs not containing a Hamiltonian path are A326224.
Cf. A000595, A002416, A003087, A003216, A283420, A326204, A326218, A326225.

A326213 Number of labeled n-vertex digraphs (with loops) not containing a (directed) Hamiltonian path.

Original entry on oeis.org

1, 2, 4, 128, 12352, 3826272, 3775441536
Offset: 0

Views

Author

Gus Wiseman, Jun 15 2019

Keywords

Comments

A path is Hamiltonian if it passes through every vertex exactly once.

Links

Crossrefs

The unlabeled case is A326224.
The case without loops is A326216.
Digraphs containing a Hamiltonian path are A326214.
Digraphs not containing a Hamiltonian cycle are A326220.
Cf. A000595, A002416, A003024, A003216, A057864, A283420, A326204, A326206, A326221.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Tuples[Range[n],2]],FindHamiltonianPath[Graph[Range[n],DirectedEdge@@@#]]=={}&]],{n,0,3}] (* Mathematica 10.2+ *)

Formula

A002416(n) = a(n) + A326214(n).

Extensions

a(5)-a(6) from Bert Dobbelaere, Jun 11 2024

A326217 Number of labeled n-vertex digraphs (without loops) containing a Hamiltonian path.

Original entry on oeis.org

0, 0, 3, 48, 3324, 929005, 1014750550, 4305572108670
Offset: 0

Views

Author

Gus Wiseman, Jun 15 2019

Keywords

Examples

			The a(3) = 48 edge-sets:
  {12,23}  {12,13,21}  {12,13,21,23}  {12,13,21,23,31}  {12,13,21,23,31,32}
  {12,31}  {12,13,23}  {12,13,21,31}  {12,13,21,23,32}
  {13,21}  {12,13,31}  {12,13,21,32}  {12,13,21,31,32}
  {13,32}  {12,13,32}  {12,13,23,31}  {12,13,23,31,32}
  {21,32}  {12,21,23}  {12,13,23,32}  {12,21,23,31,32}
  {23,31}  {12,21,31}  {12,13,31,32}  {13,21,23,31,32}
           {12,21,32}  {12,21,23,31}
           {12,23,31}  {12,21,23,32}
           {12,23,32}  {12,21,31,32}
           {12,31,32}  {12,23,31,32}
           {13,21,23}  {13,21,23,31}
           {13,21,31}  {13,21,23,32}
           {13,21,32}  {13,21,31,32}
           {13,23,31}  {13,23,31,32}
           {13,23,32}  {21,23,31,32}
           {13,31,32}
           {21,23,31}
           {21,23,32}
           {21,31,32}
           {23,31,32}

Links

Crossrefs

The undirected case is A326206.
The unlabeled undirected case is A057864.
The case with loops is A326214.
Unlabeled digraphs with a Hamiltonian path are A326221.
Digraphs (without loops) not containing a Hamiltonian path are A326216.
Digraphs (without loops) containing a Hamiltonian cycle are A326219.
Cf. A000595, A002416, A003024, A003216, A283420, A326204, A326208, A326213, A326225.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Select[Tuples[Range[n],2],UnsameQ@@#&]],FindHamiltonianPath[Graph[Range[n],DirectedEdge@@@#]]!={}&]],{n,4}] (* Mathematica 10.2+ *)

Formula

A053763(n) = a(n) + A326216(n).

Extensions

a(5)-a(7) from Bert Dobbelaere, Feb 21 2023

A326218 Number of non-Hamiltonian labeled n-vertex digraphs (without loops).

Original entry on oeis.org

1, 0, 3, 49, 2902
Offset: 0

Views

Author

Gus Wiseman, Jun 15 2019

Keywords

Comments

A digraph is Hamiltonian if it contains a directed cycle passing through every vertex exactly once.

Examples

			The a(3) = 49 edge-sets:
  {}  {12}  {12,13}  {12,13,21}  {12,13,21,23}
      {13}  {12,21}  {12,13,23}  {12,13,21,31}
      {21}  {12,23}  {12,13,31}  {12,13,23,32}
      {23}  {12,31}  {12,13,32}  {12,13,31,32}
      {31}  {12,32}  {12,21,23}  {12,21,23,32}
      {32}  {13,21}  {12,21,31}  {12,21,31,32}
            {13,23}  {12,21,32}  {13,21,23,31}
            {13,31}  {12,23,32}  {13,23,31,32}
            {13,32}  {12,31,32}  {21,23,31,32}
            {21,23}  {13,21,23}
            {21,31}  {13,21,31}
            {21,32}  {13,23,31}
            {23,31}  {13,23,32}
            {23,32}  {13,31,32}
            {31,32}  {21,23,31}
                     {21,23,32}
                     {21,31,32}
                     {23,31,32}

Links

Crossrefs

The unlabeled case is A326222.
The undirected case is A326207.
The case with loops is A326220.
Digraphs (without loops) containing a Hamiltonian cycle are A326219.
Digraphs (without loops) not containing a Hamiltonian path are A326216.
Cf. A000595, A002416, A003024, A003216, A246446, A326204, A326213, A326223, A326225.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Select[Tuples[Range[n],2],UnsameQ@@#&]],FindHamiltonianCycle[Graph[Range[n],DirectedEdge@@@#]]=={}&]],{n,4}] (* Mathematica 8.0+. Warning: Using HamiltonianGraphQ instead of FindHamiltonianCycle returns a(4) = 2896 which is incorrect *)

Formula

A053763(n) = a(n) + A326219(n).

A326219 Number of labeled n-vertex Hamiltonian digraphs (without loops).

Original entry on oeis.org

0, 1, 1, 15, 1194
Offset: 0

Views

Author

Gus Wiseman, Jun 15 2019

Keywords

Comments

A digraph is Hamiltonian if it contains a directed cycle passing through every vertex exactly once.

Examples

			The a(3) = 15 edge-sets:
  {12,23,31}  {12,13,21,32}  {12,13,21,23,31}  {12,13,21,23,31,32}
  {13,21,32}  {12,13,23,31}  {12,13,21,23,32}
              {12,21,23,31}  {12,13,21,31,32}
              {12,23,31,32}  {12,13,23,31,32}
              {13,21,23,32}  {12,21,23,31,32}
              {13,21,31,32}  {13,21,23,31,32}

Links

Crossrefs

The unlabeled case is A326225.
The undirected case is A326208 (without loops) or A326240 (with loops).
The case with loops is A326204.
Digraphs (without loops) not containing a Hamiltonian cycle are A326218.
Digraphs (without loops) containing a Hamiltonian path are A326217.
Cf. A000595, A002416, A003024, A003216, A053763, A246446, A326220, A326226.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Select[Tuples[Range[n],2],UnsameQ@@#&]],FindHamiltonianCycle[Graph[Range[n],DirectedEdge@@@#]]!={}&]],{n,0,4}] (* Mathematica 8.0+. Warning: Using HamiltonianGraphQ instead of FindHamiltonianCycle returns a(4) = 1200 which is incorrect *)

Formula

A053763(n) = a(n) + A326218(n).

A326214 Number of labeled n-vertex digraphs (with loops) containing a (directed) Hamiltonian path.

Original entry on oeis.org

0, 0, 12, 384, 53184
Offset: 0

Views

Author

Gus Wiseman, Jun 15 2019

Keywords

Comments

A path is Hamiltonian if it passes through every vertex exactly once.

Examples

			The a(2) = 12 edge-sets:
  {12}
  {21}
  {11,12}
  {11,21}
  {12,21}
  {12,22}
  {21,22}
  {11,12,21}
  {11,12,22}
  {11,21,22}
  {12,21,22}
  {11,12,21,22}

Links

Crossrefs

The unlabeled case is A326221.
The undirected case is A326206.
The case without loops is A326217.
Digraphs not containing a Hamiltonian path are A326213.
Digraphs containing a Hamiltonian cycle are A326204.
Cf. A000595, A002416, A003024, A003216, A057864, A326224, A326225, A326226.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Tuples[Range[n],2]],FindHamiltonianPath[Graph[Range[n],DirectedEdge@@@#]]!={}&]],{n,4}] (* Mathematica 10.2+ *)

Formula

A002416(n) = a(n) + A326213(n).
Showing 1-10 of 12 results. Next

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