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

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

Offset: 1

Author

Gus Wiseman, Dec 12 2019

Keywords

• nonn
• more

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).