A372171 - OEIS

%I #9 Aug 01 2024 01:21:00

%S 0,0,0,1,12,220,5460,191975,9596160,683389812,69270116040

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

%C The unlabeled version is A372174.

%F Inverse binomial transform of A372172.

%e The a(4) = 12 graphs:

%e   12,13,14,23

%e   12,13,14,24

%e   12,13,14,34

%e   12,13,23,24

%e   12,13,23,34

%e   12,14,23,24

%e   12,14,24,34

%e   12,23,24,34

%e   13,14,23,34

%e   13,14,24,34

%e   13,23,24,34

%e   14,23,24,34

%t cys[y_]:=Select[Subsets[Union@@y,{3}],MemberQ[y,{#[[1]],#[[2]]}] && MemberQ[y,{#[[1]],#[[3]]}] && MemberQ[y,{#[[2]],#[[3]]}]&];

%t Table[Length[Select[Subsets[Subsets[Range[n], {2}]],Union@@#==Range[n]&&Length[cys[#]]==1&]],{n,0,5}]

%Y Column k = 1 of A372167, unlabeled A372173.

%Y For no triangles we have A372168 (non-covering A213434), unlabeled A372169.

%Y The non-covering case is A372172, unlabeled A372194.

%Y The unlabeled version is A372174.

%Y For all cycles (not just triangles) we have A372195, non-covering A372193.

%Y A001858 counts acyclic graphs, unlabeled A005195.

%Y A006125 counts simple graphs, unlabeled A000088.

%Y A006129 counts covering graphs, unlabeled A002494

%Y A054548 counts labeled covering graphs by edges, unlabeled A370167.

%Y A105784 counts acyclic covering graphs, unlabeled A144958.

%Y A372170 counts graphs by triangles, unlabeled A263340.

%Y A372175 counts covering graphs by cycles, non-covering A372176.

%Y Cf. A000272, A053530, A121251, A137916, A367868, A369199, A372191.

%K nonn,more

%O 0,5

%A _Gus Wiseman_, Apr 24 2024

%E a(7)-a(10) from _Andrew Howroyd_, Aug 01 2024