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A327357 Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of antichains of sets covering n vertices with non-spanning edge-connectivity k.

%I A327357 #4 Sep 11 2019 20:22:21
%S A327357 1,0,1,1,1,4,1,3,1,30,13,33,32,6,546,421,1302,1915,1510,693,316,135,
%T A327357 45,10,1
%N A327357 Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of antichains of sets covering n vertices with non-spanning edge-connectivity k.
%C A327357 An antichain is a set of sets, none of which is a subset of any other. It is covering if there are no isolated vertices.
%C A327357 The non-spanning edge-connectivity of a set-system is the minimum number of edges that must be removed (along with any non-covered vertices) to obtain a disconnected or empty set-system.
%e A327357 Triangle begins:
%e A327357     1
%e A327357     0    1
%e A327357     1    1
%e A327357     4    1    3    1
%e A327357    30   13   33   32    6
%e A327357   546  421 1302 1915 1510  693  316  135   45   10    1
%e A327357 Row n = 3 counts the following antichains:
%e A327357   {{1},{2,3}}    {{1,2,3}}  {{1,2},{1,3}}  {{1,2},{1,3},{2,3}}
%e A327357   {{2},{1,3}}               {{1,2},{2,3}}
%e A327357   {{3},{1,2}}               {{1,3},{2,3}}
%e A327357   {{1},{2},{3}}
%t A327357 csm[s_]:=With[{c=Select[Subsets[Range[Length[s]],{2}],Length[Intersection@@s[[#]]]>0&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
%t A327357 stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
%t A327357 eConn[sys_]:=If[Length[csm[sys]]!=1,0,Length[sys]-Max@@Length/@Select[Union[Subsets[sys]],Length[csm[#]]!=1&]];
%t A327357 Table[Length[Select[stableSets[Subsets[Range[n],{1,n}],SubsetQ],Union@@#==Range[n]&&eConn[#]==k&]],{n,0,5},{k,0,2^n}]//.{foe___,0}:>{foe}
%Y A327357 Row sums are A307249.
%Y A327357 Column k = 0 is A120338.
%Y A327357 The non-covering version is A327353.
%Y A327357 The version for spanning edge-connectivity is A327352.
%Y A327357 The specialization to simple graphs is A327149, with unlabeled version A327201.
%Y A327357 Cf. A014466, A293606, A326704, A326787, A327071, A327148, A327236, A327351.
%K A327357 nonn,tabf,more
%O A327357 0,6
%A A327357 _Gus Wiseman_, Sep 11 2019