
MATHEMATICA

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]]]]]]]]];
spanEdgeConn[vts_, eds_]:=Length[eds]Max@@Length/@Select[Subsets[eds], Union@@#!=vtsLength[csm[#]]!=1&];
Table[Length[Select[Subsets[Subsets[Range[n], {1, n}]], spanEdgeConn[Range[n], #]==1&]], {n, 0, 3}]


CROSSREFS

The BIInumbers of these setsystems are A327111.
Set systems with nonspanning edgeconnectivity 1 are A327196, with covering case A327129.
Set systems with spanning edgeconnectivity 2 are A327130.
Cf. A003465, A323818, A327069, A327071, A327099, A327108, A327144.
Sequence in context: A158262 A300513 A089035 * A089516 A000573 A070019
Adjacent sequences: A327142 A327143 A327144 * A327146 A327147 A327148
