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 1-6 of 6 results.

A326204 Number of Hamiltonian labeled n-vertex digraphs (with loops).

Original entry on oeis.org

0, 2, 4, 120, 19104
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

			The a(2) = 4 digraph edge-sets:
  {12,21}
  {11,12,21}
  {12,21,22}
  {11,12,21,22}

Links

Crossrefs

The unlabeled case is A326226.
The case without loops is A326219.
The undirected case (without loops) is A326208.
Non-Hamiltonian digraphs are A326220.
Digraphs containing a Hamiltonian path are A326214.
Cf. A000595, A002416, A003024, A003216, A057864.

Programs

  • Mathematica
    Table[Length[Select[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) = 19200 which is incorrect *)

A326206 Number of n-vertex labeled simple graphs containing a Hamiltonian path.

Original entry on oeis.org

0, 0, 1, 4, 34, 633, 23368, 1699012, 237934760, 64558137140, 34126032806936, 35513501049012952
Offset: 0

Views

Author

Gus Wiseman, Jun 14 2019

Keywords

Comments

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

Links

Crossrefs

The unlabeled case is A057864.
The directed case is A326214 (with loops) or A326217 (without loops).
Simple graphs without a Hamiltonian path are A326205.
Simple graphs with a Hamiltonian cycle are A326208.
Cf. A003216, A006125, A057864, A283420.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Subsets[Range[n],{2}]],FindHamiltonianPath[Graph[Range[n],#]]!={}&]],{n,0,4}] (* Mathematica 10.2+ *)

Formula

A006125(n) = a(n) + A326205(n).

Extensions

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

A326221 Number of unlabeled n-vertex digraphs (with loops) containing a Hamiltonian path.

Original entry on oeis.org

0, 0, 7, 74, 2395
Offset: 0

Views

Author

Gus Wiseman, Jun 16 2019

Keywords

Comments

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

Links

Crossrefs

The labeled case is A326214.
The undirected case is A057864 (without loops).
Unlabeled digraphs not containing a Hamiltonian path are A326224.
Unlabeled digraphs containing a Hamiltonian cycle are A326226.
Cf. A000595, A002416, A003087, A003216, A283420.

Formula

A000595(n) = a(n) + A326224(n).

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

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).
Showing 1-6 of 6 results.

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

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