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A330297
Number of labeled simple graphs covering n vertices with exactly two automorphisms, or with exactly n!/2 graphs obtainable by permuting the vertices.
5
0, 0, 1, 3, 24, 540, 13320
OFFSET
0,4
COMMENTS
These are graphs with exactly one involution and no other symmetries.
FORMULA
a(n) = n!/2 * A330346(n).
EXAMPLE
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}
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}]
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).
Sequence in context: A336577 A194157 A166736 * A109055 A318766 A292813
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Dec 12 2019
STATUS
approved