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 1-5 of 5 results.

A330282 Number of fully chiral set-systems on n vertices.

Original entry on oeis.org

1, 2, 5, 52, 21521
Offset: 0

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Author

Gus Wiseman, Dec 10 2019

Keywords

Comments

A set-system is a finite set of finite nonempty sets. It is fully chiral if every permutation of the covered vertices gives a different representative.

Examples

			The a(0) = 1 through a(2) = 5 set-systems:
  {}  {}     {}
      {{1}}  {{1}}
             {{2}}
             {{1},{1,2}}
             {{2},{1,2}}

Crossrefs

Costrict (or T_0) set-systems are A326940.
The covering case is A330229.
The unlabeled version is A330294, with covering case A330295.
Achiral set-systems are A083323.
BII-numbers of fully chiral set-systems are A330226.
Non-isomorphic fully chiral multiset partitions are A330227.
Fully chiral partitions are A330228.
Fully chiral factorizations are A330235.
MM-numbers of fully chiral multisets of multisets are A330236.
Cf. A000612, A016031, A319637, A330098, A330231, A330232, A330234.

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],{1,n}]],Length[graprms[#]]==Length[Union@@#]!&]],{n,0,3}]

Formula

Binomial transform of A330229.

A330295 Number of non-isomorphic fully chiral set-systems covering n vertices.

Original entry on oeis.org

1, 1, 1, 7, 889
Offset: 0

Views

Author

Gus Wiseman, Dec 10 2019

Keywords

Comments

A set-system is a finite set of finite nonempty sets. It is fully chiral if every permutation of the covered vertices gives a different representative.

Examples

			Non-isomorphic representatives of the a(0) = 1 through a(3) = 7 set-systems:
  0  {1}  {1}{12}  {1}{2}{13}
                   {1}{12}{23}
                   {1}{12}{123}
                   {1}{2}{12}{13}
                   {1}{2}{13}{123}
                   {1}{12}{23}{123}
                   {1}{2}{12}{13}{123}

Crossrefs

The labeled version is A330229.
First differences of A330294 (the non-covering case).
Unlabeled costrict (or T_0) set-systems are A326946.
BII-numbers of fully chiral set-systems are A330226.
Non-isomorphic fully chiral multiset partitions are A330227.
Fully chiral partitions are A330228.
Fully chiral factorizations are A330235.
MM-numbers of fully chiral multisets of multisets are A330236.
Cf. A000612, A016031, A055621, A083323, A283877, A319637, A330098, A330231, A330232, A330234, A330282.

A330344 Number of unlabeled graphs with n vertices whose covered portion has exactly two automorphisms.

Original entry on oeis.org

0, 1, 2, 4, 13, 50, 367
Offset: 1

Views

Author

Gus Wiseman, Dec 12 2019

Keywords

Examples

			Non-isomorphic representatives of the a(2) = 1 through a(5) = 13 graphs:
  {12}  {12}     {12}           {12}
        {12,13}  {12,13}        {12,13}
                 {12,13,24}     {12,13,24}
                 {12,13,14,23}  {12,13,14,23}
                                {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}

Crossrefs

The labeled version is A330345.
The covering case is A330346 (not A241454).
Unlabeled graphs are A000088.
Unlabeled graphs with exactly one automorphism are A003400.
Unlabeled connected graphs with exactly one automorphism are A124059.
Graphs with exactly two automorphisms are A330297 (labeled covering), A330344 (unlabeled), A330345 (labeled), and A330346 (unlabeled covering).
Cf. A000612, A004111, A055621, A241454, A283877, A330098, A330227, A330230, A330231, A330233, A330294, A330295.

Formula

Partial sums of A330346.

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.

A330343 Number of labeled fully chiral simple graphs (also called identity or asymmetric graphs) covering n vertices.

Original entry on oeis.org

1, 0, 0, 0, 0, 5760, 766080, 149022720, 48990251520, 28928242022400, 32147584690636800, 69035206021583155200
Offset: 1

Views

Author

Gus Wiseman, Dec 12 2019

Keywords

Comments

In a fully chiral graph, every permutation of the vertices gives a different representative, so the only automorphism is the identity.

Crossrefs

The unlabeled version is A003400.
Identity trees are A004111.
Covering simple graphs are A006129.
Full chiral integer partitions are A330228.
Fully chiral factorizations are A330235.
Fully chiral set-systems are A330229 (labeled covering), A330282 (labeled), A330294 (unlabeled), A330295 (unlabeled covering).
Graphs with exactly two automorphisms are A330297 (labeled covering), A330344 (unlabeled), A330345 (labeled), A330346 (unlabeled covering), A241454 (unlabeled connected).
Cf. A006125, A016031, A124059, A143543, A330098, A330224, A330226, A330227, A330230, A330231, A330236.

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[#]]==n!&]],{n,5}] (* brute force, not for computation *)

Formula

a(n) = n! * A003400(n).
Showing 1-5 of 5 results.

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