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%I #14 May 26 2021 02:14:33
%S 1,1,0,1,1,0,4,3,1,0,26,28,9,1,0,296,475,227,25,1,0,6064,14736,10110,
%T 1782,75,1,0
%N Triangle read by rows where T(n,k) is the number of labeled simple graphs with n vertices and spanning edge-connectivity k.
%C The spanning edge-connectivity of a graph is the minimum number of edges that must be removed (without removing incident vertices) to obtain a disconnected or empty graph.
%C We consider a graph with one vertex and no edges to be disconnected.
%e Triangle begins:
%e 1
%e 1 0
%e 1 1 0
%e 4 3 1 0
%e 26 28 9 1 0
%e 296 475 227 25 1 0
%t 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 spanEdgeConn[vts_,eds_]:=Length[eds]-Max@@Length/@Select[Subsets[eds],Union@@#!=vts||Length[csm[#]]!=1&];
%t Table[Length[Select[Subsets[Subsets[Range[n],{2}]],spanEdgeConn[Range[n],#]==k&]],{n,0,5},{k,0,n}]
%Y Row sums are A006125.
%Y Column k = 0 is A054592, if we assume A054592(1) = 1.
%Y Column k = 1 is A327071.
%Y Column k = 2 is A327146.
%Y The unlabeled version (except with offset 1) is A263296.
%Y Cf. A001187, A095983, A259862, A322338, A326787, A327070, A327072, A327073.
%K nonn,tabl,more
%O 0,7
%A _Gus Wiseman_, Aug 23 2019
%E a(21)-a(27) from _Robert Price_, May 25 2021
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