A330345 - OEIS

%I #6 Dec 12 2019 09:33:37

%S 0,0,1,6,42,700,16995

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

%e The a(4) = 42 graphs:

%e   {12}  {12,13}  {12,13,24}  {12,13,14,23}

%e   {13}  {12,14}  {12,13,34}  {12,13,14,24}

%e   {14}  {12,23}  {12,14,23}  {12,13,14,34}

%e   {23}  {12,24}  {12,14,34}  {12,13,23,24}

%e   {24}  {13,14}  {12,23,34}  {12,13,23,34}

%e   {34}  {13,23}  {12,24,34}  {12,14,23,24}

%e         {13,34}  {13,14,23}  {12,14,24,34}

%e         {14,24}  {13,14,24}  {12,23,24,34}

%e         {14,34}  {13,23,24}  {13,14,23,34}

%e         {23,24}  {13,24,34}  {13,14,24,34}

%e         {23,34}  {14,23,24}  {13,23,24,34}

%e         {24,34}  {14,23,34}  {14,23,24,34}

%t graprms[m_]:=Union[Table[Sort[Sort/@(m/.Rule@@@Table[{p[[i]],i},{i,Length[p]}])],{p,Permutations[Union@@m]}]];

%t Table[Length[Select[Subsets[Subsets[Range[n],{2}]],Length[graprms[#]]==Length[Union@@#]!/2&]],{n,0,4}]

%Y The unlabeled version is A330344.

%Y The covering case is A330297.

%Y Covering simple graphs are A006129.

%Y Graphs with exactly two automorphisms are A330297 (labeled covering), A330344 (unlabeled), A330345 (labeled), A330346 (unlabeled covering), A241454 (unlabeled connected).

%Y Cf. A006125, A143543, A330098, A330228, A330230, A330282, A330343.

%K nonn,more

%O 0,4

%A _Gus Wiseman_, Dec 12 2019