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EXAMPLE
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Non-isomorphic representatives of the a(3) = 8 set-systems:
0,
{12}, {123},
{12}{13}, {12}{123},
{12}{13}{23}, {12}{13}{123},
{12}{13}{23}{123}.
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MATHEMATICA
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sysnorm[{}] := {}; sysnorm[m_]:=If[Union@@m!=Range[Max@@Flatten[m]], sysnorm[m/.Rule@@@Table[{(Union@@m)[[i]], i}, {i, Length[Union@@m]}]], First[Sort[sysnorm[m, 1]]]]; sysnorm[m_, aft_]:=If[Length[Union@@m]<=aft, {m}, With[{mx=Table[Count[m, i, {2}], {i, Select[Union@@m, #>=aft&]}]}, Union@@(sysnorm[#, aft+1]&/@Union[Table[Map[Sort, m/.{par+aft-1->aft, aft->par+aft-1}, {0, 1}], {par, First/@Position[mx, Max[mx]]}]])]];
Table[Length[Union[sysnorm/@Select[Subsets[Select[Subsets[Range[n]], Length[#]>1&]], Or[Length[#]==0, Union@@#==Range[Max@@Union@@#]]&]]], {n, 4}]
(* second program *)
Table[Sum[2^PermutationCycles[Ordering[Map[Sort, Subsets[Range[n], {2, n}]/.Rule@@@Table[{i, prm[[i]]}, {i, n}], {1}]], Length]/n!, {prm, Permutations[Range[n]]}], {n, 6}] (* Gus Wiseman, Dec 12 2018 *)
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