cp's OEIS Frontend

A372171 Number of labeled simple graphs covering n vertices with a unique triangle.

%I A372171 #9 Aug 01 2024 01:21:00
%S A372171 0,0,0,1,12,220,5460,191975,9596160,683389812,69270116040
%N A372171 Number of labeled simple graphs covering n vertices with a unique triangle.
%C A372171 The unlabeled version is A372174.
%F A372171 Inverse binomial transform of A372172.
%e A372171 The a(4) = 12 graphs:
%e A372171   12,13,14,23
%e A372171   12,13,14,24
%e A372171   12,13,14,34
%e A372171   12,13,23,24
%e A372171   12,13,23,34
%e A372171   12,14,23,24
%e A372171   12,14,24,34
%e A372171   12,23,24,34
%e A372171   13,14,23,34
%e A372171   13,14,24,34
%e A372171   13,23,24,34
%e A372171   14,23,24,34
%t A372171 cys[y_]:=Select[Subsets[Union@@y,{3}],MemberQ[y,{#[[1]],#[[2]]}] && MemberQ[y,{#[[1]],#[[3]]}] && MemberQ[y,{#[[2]],#[[3]]}]&];
%t A372171 Table[Length[Select[Subsets[Subsets[Range[n], {2}]],Union@@#==Range[n]&&Length[cys[#]]==1&]],{n,0,5}]
%Y A372171 Column k = 1 of A372167, unlabeled A372173.
%Y A372171 For no triangles we have A372168 (non-covering A213434), unlabeled A372169.
%Y A372171 The non-covering case is A372172, unlabeled A372194.
%Y A372171 The unlabeled version is A372174.
%Y A372171 For all cycles (not just triangles) we have A372195, non-covering A372193.
%Y A372171 A001858 counts acyclic graphs, unlabeled A005195.
%Y A372171 A006125 counts simple graphs, unlabeled A000088.
%Y A372171 A006129 counts covering graphs, unlabeled A002494
%Y A372171 A054548 counts labeled covering graphs by edges, unlabeled A370167.
%Y A372171 A105784 counts acyclic covering graphs, unlabeled A144958.
%Y A372171 A372170 counts graphs by triangles, unlabeled A263340.
%Y A372171 A372175 counts covering graphs by cycles, non-covering A372176.
%Y A372171 Cf. A000272, A053530, A121251, A137916, A367868, A369199, A372191.
%K A372171 nonn,more
%O A372171 0,5
%A A372171 _Gus Wiseman_, Apr 24 2024
%E A372171 a(7)-a(10) from _Andrew Howroyd_, Aug 01 2024