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%I #14 May 26 2021 02:14:43
%S 1,1,1,1,1,3,3,1,4,18,27,14,1,56,250,402,240,65,10,1,1031,5475,11277,
%T 9620,4282,921,146,15,1
%N Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of labeled simple graphs with n vertices and non-spanning edge-connectivity k.
%C The non-spanning edge-connectivity of a graph is the minimum number of edges that must be removed (along with any isolated vertices) to obtain a disconnected or empty graph.
%F T(n,k) = Sum_{m = 0..n} binomial(n,m) A327149(m,k). In words, column k is the binomial transform of column k of A327149.
%e Triangle begins:
%e 1
%e 1
%e 1 1
%e 1 3 3 1
%e 4 18 27 14 1
%e 56 250 402 240 65 10 1
%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 edgeConnSys[sys_]:=If[Length[csm[sys]]!=1,0,Length[sys]-Max@@Length/@Select[Union[Subsets[sys]],Length[csm[#]]!=1&]];
%t Table[Length[Select[Subsets[Subsets[Range[n],{2}]],edgeConnSys[#]==k&]],{n,0,4},{k,0,Binomial[n,2]}]//.{foe___,0}:>{foe}
%Y Row sums are A006125.
%Y Column k = 0 is A327199.
%Y Column k = 1 is A327231.
%Y The corresponding triangle for vertex-connectivity is A327125.
%Y The corresponding triangle for spanning edge-connectivity is A327069.
%Y The covering version is A327149.
%Y The unlabeled version is A327236, with covering version A327201.
%Y Cf. A001187, A263296, A322338, A322395, A326787, A327079, A327097, A327099, A327102, A327126, A327144, A327196, A327200, A327201.
%K nonn,tabf,more
%O 0,6
%A _Gus Wiseman_, Aug 27 2019
%E a(20)-a(28) from _Robert Price_, May 25 2021
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