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.

A330297 Number of labeled simple graphs covering n vertices with exactly two automorphisms, or with exactly n!/2 graphs obtainable by permuting the vertices.

Original entry on oeis.org

0, 0, 1, 3, 24, 540, 13320
Offset: 0

Views

Author

Gus Wiseman, Dec 12 2019

Keywords

Comments

These are graphs with exactly one involution and no other symmetries.

Examples

			The a(4) = 24 graphs:
  {12,13,24}  {12,13,14,23}
  {12,13,34}  {12,13,14,24}
  {12,14,23}  {12,13,14,34}
  {12,14,34}  {12,13,23,24}
  {12,23,34}  {12,13,23,34}
  {12,24,34}  {12,14,23,24}
  {13,14,23}  {12,14,24,34}
  {13,14,24}  {12,23,24,34}
  {13,23,24}  {13,14,23,34}
  {13,24,34}  {13,14,24,34}
  {14,23,24}  {13,23,24,34}
  {14,23,34}  {14,23,24,34}

Links

Crossrefs

The non-covering version is A330345.
The unlabeled version is A330346 (not A241454).
Covering simple graphs are A006129.
Covering graphs with exactly one automorphism are A330343.
Graphs with exactly two automorphisms are A330297 (labeled covering), A330344 (unlabeled), A330345 (labeled), and A330346 (unlabeled covering).
Cf. A003400, A006125, A016031, A124059, A143543, A241454, A330098, A330229, A330230, A330231, A330233.

Programs

  • Mathematica
    graprms[m_]:=Union[Table[Sort[Sort/@(m/.Rule@@@Table[{p[[i]],i},{i,Length[p]}])],{p,Permutations[Union@@m]}]];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],Union@@#==Range[n]&&Length[graprms[#]]==n!/2&]],{n,0,5}]

Formula

a(n) = n!/2 * A330346(n).

A330345 Number of labeled simple graphs with n vertices whose covered portion has exactly two automorphisms.

Original entry on oeis.org

0, 0, 1, 6, 42, 700, 16995
Offset: 0

Views

Author

Gus Wiseman, Dec 12 2019

Keywords

Examples

			The a(4) = 42 graphs:
  {12}  {12,13}  {12,13,24}  {12,13,14,23}
  {13}  {12,14}  {12,13,34}  {12,13,14,24}
  {14}  {12,23}  {12,14,23}  {12,13,14,34}
  {23}  {12,24}  {12,14,34}  {12,13,23,24}
  {24}  {13,14}  {12,23,34}  {12,13,23,34}
  {34}  {13,23}  {12,24,34}  {12,14,23,24}
        {13,34}  {13,14,23}  {12,14,24,34}
        {14,24}  {13,14,24}  {12,23,24,34}
        {14,34}  {13,23,24}  {13,14,23,34}
        {23,24}  {13,24,34}  {13,14,24,34}
        {23,34}  {14,23,24}  {13,23,24,34}
        {24,34}  {14,23,34}  {14,23,24,34}

Crossrefs

The unlabeled version is A330344.
The covering case is A330297.
Covering simple graphs are A006129.
Graphs with exactly two automorphisms are A330297 (labeled covering), A330344 (unlabeled), A330345 (labeled), A330346 (unlabeled covering), A241454 (unlabeled connected).
Cf. A006125, A143543, A330098, A330228, A330230, A330282, A330343.

Programs

  • Mathematica
    graprms[m_]:=Union[Table[Sort[Sort/@(m/.Rule@@@Table[{p[[i]],i},{i,Length[p]}])],{p,Permutations[Union@@m]}]];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],Length[graprms[#]]==Length[Union@@#]!/2&]],{n,0,4}]

A330346 Number of unlabeled simple graphs covering n vertices with exactly two automorphisms.

Original entry on oeis.org

0, 0, 1, 1, 2, 9, 37
Offset: 0

Views

Author

Gus Wiseman, Dec 12 2019

Keywords

Comments

First differs from A241454 at n = 8.

Examples

			Non-isomorphic representatives of the a(5) = 9 graphs:
  {12,13,14,25}
  {12,13,24,35}
  {12,13,14,23,25}
  {12,13,14,23,45}
  {12,13,15,24,34}
  {12,13,14,15,23,24}
  {12,13,14,23,24,35}
  {12,13,14,23,25,45}
  {12,13,14,15,23,24,35}

Links

Crossrefs

The labeled version is A330297.
The non-covering version is A330344.
Unlabeled covering graphs are A002494.
Unlabeled connected graphs with exactly two automorphisms are A241454.
Graphs with exactly two automorphisms are A330297 (labeled covering), A330344 (unlabeled), A330345 (labeled), A330346 (unlabeled covering), A241454 (unlabeled connected).
Cf. A000088, A003400, A055621, A124059, A330098, A330227, A330230, A330231, A330294, A330295, A330343.
Showing 1-3 of 3 results.

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