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%I #10 Jan 11 2024 13:53:40
%S 1,1,2,10,97,2160,126862,21485262,11105374322,18109358131513,
%T 95465831661532570,1660400673336788987026,96929369602251313489896310,
%U 19268528295096123543660356281600,13203875101002459910158494602665950757,31517691852305548841992346407978317698725021
%N Number of non-isomorphic n-element sets of nonempty subsets of {1..n}.
%H Andrew Howroyd, <a href="/A368731/b368731.txt">Table of n, a(n) for n = 0..50</a>
%e Non-isomorphic representatives of the a(3) = 10 set-systems:
%e {{1},{2},{3}}
%e {{1},{2},{1,2}}
%e {{1},{2},{1,3}}
%e {{1},{2},{1,2,3}}
%e {{1},{1,2},{1,3}}
%e {{1},{1,2},{2,3}}
%e {{1},{1,2},{1,2,3}}
%e {{1},{2,3},{1,2,3}}
%e {{1,2},{1,3},{2,3}}
%e {{1,2},{1,3},{1,2,3}}
%t brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{(Union@@m)[[i]],p[[i]]},{i,Length[p]}])], {p,Permutations[Range[Length[Union@@m]]]}]]];
%t Table[Length[Union[brute /@ Subsets[Subsets[Range[n],{1,n}],{n}]]],{n,0,4}]
%o (PARI) a(n) = polcoef(G(n, n), n) \\ G defined in A368186. - _Andrew Howroyd_, Jan 11 2024
%Y The case of graphs is A001434, labeled A116508.
%Y Labeled version is A136556, covering A054780, binomial transform of A367916.
%Y The case of labeled covering graphs is A367863, binomial transform A367862.
%Y These include the set-systems ranked by A367917.
%Y The covering case is A368186, for graphs A006649, connected A057500.
%Y Requiring all edges to be singletons or pairs gives A368598.
%Y A003465 counts covers with any number of edges, unlabeled A055621.
%Y A046165 counts minimal covers, ranks A309326.
%Y A058891 counts set-systems, unlabeled A000612, without singletons A016031.
%Y A059201 counts covering T_0 set-systems, unlabeled A319637, ranks A326947.
%Y Cf. A000088, A007716, A283877, A367901, A367902, A367903, A368599.
%K nonn
%O 0,3
%A _Gus Wiseman_, Jan 07 2024
%E Terms a(6) and beyond from _Andrew Howroyd_, Jan 11 2024
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