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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.

%I A330297 #10 Jan 05 2020 12:03:24
%S A330297 0,0,1,3,24,540,13320
%N A330297 Number of labeled simple graphs covering n vertices with exactly two automorphisms, or with exactly n!/2 graphs obtainable by permuting the vertices.
%C A330297 These are graphs with exactly one involution and no other symmetries.
%H A330297 Gus Wiseman, <a href="/A330297/a330297.png">All 9 distinct unlabeled representatives of the a(5) = 540 graphs.</a>
%F A330297 a(n) = n!/2 * A330346(n).
%e A330297 The a(4) = 24 graphs:
%e A330297   {12,13,24}  {12,13,14,23}
%e A330297   {12,13,34}  {12,13,14,24}
%e A330297   {12,14,23}  {12,13,14,34}
%e A330297   {12,14,34}  {12,13,23,24}
%e A330297   {12,23,34}  {12,13,23,34}
%e A330297   {12,24,34}  {12,14,23,24}
%e A330297   {13,14,23}  {12,14,24,34}
%e A330297   {13,14,24}  {12,23,24,34}
%e A330297   {13,23,24}  {13,14,23,34}
%e A330297   {13,24,34}  {13,14,24,34}
%e A330297   {14,23,24}  {13,23,24,34}
%e A330297   {14,23,34}  {14,23,24,34}
%t A330297 graprms[m_]:=Union[Table[Sort[Sort/@(m/.Rule@@@Table[{p[[i]],i},{i,Length[p]}])],{p,Permutations[Union@@m]}]];
%t A330297 Table[Length[Select[Subsets[Subsets[Range[n],{2}]],Union@@#==Range[n]&&Length[graprms[#]]==n!/2&]],{n,0,5}]
%Y A330297 The non-covering version is A330345.
%Y A330297 The unlabeled version is A330346 (not A241454).
%Y A330297 Covering simple graphs are A006129.
%Y A330297 Covering graphs with exactly one automorphism are A330343.
%Y A330297 Graphs with exactly two automorphisms are A330297 (labeled covering), A330344 (unlabeled), A330345 (labeled), and A330346 (unlabeled covering).
%Y A330297 Cf. A003400, A006125, A016031, A124059, A143543, A241454, A330098, A330229, A330230, A330231, A330233.
%K A330297 nonn,more
%O A330297 0,4
%A A330297 _Gus Wiseman_, Dec 12 2019