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%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
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