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MATHEMATICA
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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]]]]]]]]];
vConn[sys_]:=If[Length[csm[sys]]!=1, 0, Min@@Length/@Select[Subsets[Union@@sys], Function[del, Length[csm[DeleteCases[DeleteCases[sys, Alternatives@@del, {2}], {}]]]!=1]]];
Table[Length[Select[Subsets[Subsets[Range[n], {1, n}]], Union@@#==Range[n]&&vConn[#]>=2&]], {n, 0, 3}]
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CROSSREFS
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Covering 2-cut-connected graphs are A013922, if we assume A013922(2) = 1.
Covering 2-cut-connected antichains (blobs) are A275307, if we assume A275307(1) = 0.
Covering set-systems with cut-connectivity 2 are A327113.
2-vertex-connected integer partitions are A322387.
BII-numbers of set-systems with cut-connectivity >= 2 are A327101.
The cut-connectivity of the set-system with BII-number n is A326786(n).
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