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 1-7 of 7 results.

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

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.

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).

A326216 Number of labeled n-vertex digraphs (without loops) not containing a (directed) Hamiltonian path.

Original entry on oeis.org

1, 1, 1, 16, 772
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(3) = 16 edge-sets:
  {}  {12}  {12,13}
      {13}  {12,21}
      {21}  {12,32}
      {23}  {13,23}
      {31}  {13,31}
      {32}  {21,23}
            {21,31}
            {23,32}
            {31,32}

Links

Crossrefs

Unlabeled digraphs not containing a Hamiltonian path are A326224.
The undirected case is A326205.
The unlabeled undirected case is A283420.
The case with loops is A326213.
Digraphs (without loops) containing a Hamiltonian path are A326217.
Digraphs (without loops) not containing a Hamiltonian cycle are A326218.
Cf. A000595, A002416, A003024, A003216, A057864, A326206, A326214, A326220, A326221, 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) + A326217(n).

A326222 Number of non-Hamiltonian unlabeled n-vertex digraphs (without loops).

Original entry on oeis.org

1, 0, 2, 12, 157, 5883, 696803, 255954536
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.

Links

Crossrefs

The labeled case is A326218 (without loops) or A326220 (with loops).
The undirected case (without loops) is A246446.
The case with loops is A326223.
Hamiltonian unlabeled digraphs are A326225 (without loops) or A003216 (with loops).
Cf. A000273, A000595, A002416, A003087, A053763, A326216, A326217, A326224, A326226.

Formula

a(n) = A000273(n) - A326225(n). - Pontus von Brömssen, Mar 17 2024

Extensions

a(5)-a(7) (using A000273 and A326225) from Pontus von Brömssen, Mar 17 2024

A326239 Number of non-Hamiltonian labeled n-vertex graphs with loops.

Original entry on oeis.org

1, 0, 8, 56, 864, 25792
Offset: 0

Views

Author

Gus Wiseman, Jun 16 2019

Keywords

Comments

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

Examples

			The a(3) = 56 edge-sets:
  {}  {11}  {11,12}  {11,12,13}
      {12}  {11,13}  {11,12,22}
      {13}  {11,22}  {11,12,23}
      {22}  {11,23}  {11,12,33}
      {23}  {11,33}  {11,13,22}
      {33}  {12,13}  {11,13,23}
            {12,22}  {11,13,33}
            {12,23}  {11,22,23}
            {12,33}  {11,22,33}
            {13,22}  {11,23,33}
            {13,23}  {12,13,22}
            {13,33}  {12,13,33}
            {22,23}  {12,22,23}
            {22,33}  {12,22,33}
            {23,33}  {12,23,33}
                     {13,22,23}
                     {13,22,33}
                     {13,23,33}
                     {22,23,33}

Links

Crossrefs

The directed case is A326204 (with loops) or A326218 (without loops).
Simple graphs containing a Hamiltonian cycle are A326240.
Simple graphs not containing a Hamiltonian path are A326205.
Cf. A000088, A003216, A006125, A057864, A283420.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Select[Tuples[Range[n],2],OrderedQ]],FindHamiltonianCycle[Graph[Range[n],#]]=={}&]],{n,0,4}]
Showing 1-7 of 7 results.

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