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EXAMPLE
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The a(3) = 35 set-systems:
{123} {1}{12}{23} {1}{2}{12}{13} {1}{2}{3}{12}{13}
{1}{13}{23} {1}{2}{12}{23} {1}{2}{3}{12}{23}
{1}{2}{123} {1}{2}{13}{23} {1}{2}{3}{13}{23}
{1}{3}{123} {1}{2}{3}{123} {1}{2}{3}{12}{123}
{2}{12}{13} {1}{3}{12}{13} {1}{2}{3}{13}{123}
{2}{13}{23} {1}{3}{12}{23} {1}{2}{3}{23}{123}
{2}{3}{123} {1}{3}{13}{23}
{3}{12}{13} {2}{3}{12}{13}
{3}{12}{23} {2}{3}{12}{23}
{1}{23}{123} {2}{3}{13}{23}
{2}{13}{123} {1}{2}{13}{123}
{3}{12}{123} {1}{2}{23}{123}
{1}{3}{12}{123}
{1}{3}{23}{123}
{2}{3}{12}{123}
{2}{3}{13}{123}
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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]]]]]]]]];
eConn[sys_]:=If[Length[csm[sys]]!=1, 0, Length[sys]-Max@@Length/@Select[Union[Subsets[sys]], Length[csm[#]]!=1&]];
Table[Length[Select[Subsets[Subsets[Range[n], {1, n}]], Union@@#==Range[n]&&eConn[#]==1&]], {n, 0, 3}]
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