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A327126 Triangle read by rows where T(n,k) is the number of labeled simple graphs covering n vertices with cut-connectivity k.

%I A327126 #17 May 20 2021 22:59:12
%S A327126 1,0,0,0,0,1,0,3,0,1,3,28,9,0,1,40,490,212,25,0,1,745,15336,9600,1692,
%T A327126 75,0,1
%N A327126 Triangle read by rows where T(n,k) is the number of labeled simple graphs covering n vertices with cut-connectivity k.
%C A327126 We define the cut-connectivity of a graph to be the minimum number of vertices that must be removed (along with any incident edges) to obtain a disconnected or empty graph, with the exception that a graph with one vertex and no edges has cut-connectivity 1. Except for complete graphs, this is the same as vertex-connectivity.
%e A327126 Triangle begins:
%e A327126    1
%e A327126    0   0
%e A327126    0   0   1
%e A327126    0   3   0   1
%e A327126    3  28   9   0   1
%e A327126   40 490 212  25   0   1
%t A327126 csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
%t A327126 cutConnSys[vts_,eds_]:=If[Length[vts]==1,1,Min@@Length/@Select[Subsets[vts],Function[del,csm[DeleteCases[DeleteCases[eds,Alternatives@@del,{2}],{}]]!={Complement[vts,del]}]]];
%t A327126 Table[Length[Select[Subsets[Subsets[Range[n],{2}]],Union@@#==Range[n]&&cutConnSys[Range[n],#]==k&]],{n,0,4},{k,0,n}]
%Y A327126 After the first column, same as A327125.
%Y A327126 Column k = 0 is A327070.
%Y A327126 Column k = 1 is A327114.
%Y A327126 Row sums are A006129.
%Y A327126 Different from A327069.
%Y A327126 Row sums without the first column are A001187, if we assume A001187(0) = A001187(1) = 0.
%Y A327126 Row sums without the first two columns are A013922.
%Y A327126 Cf. A006125, A259862, A322389, A326786, A327114, A327127, A327198, A327237.
%K A327126 nonn,more,tabl
%O A327126 0,8
%A A327126 _Gus Wiseman_, Aug 25 2019
%E A327126 a(21)-a(27) from _Robert Price_, May 20 2021