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%I #13 Jan 29 2024 13:48:25
%S 1,1,3,86,31302,2146841520,9223371978880250448,
%T 170141183460469231408869283342774399392,
%U 57896044618658097711785492504343953919148780260559635830120038252613826101856
%N Number of connected T_0 set-systems on n vertices.
%C The dual of a set-system has, for each vertex, one edge consisting of the indices (or positions) of the edges containing that vertex. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. The T_0 condition means that the dual is strict (no repeated edges).
%H Andrew Howroyd, <a href="/A326948/b326948.txt">Table of n, a(n) for n = 0..11</a>
%F Logarithmic transform of A059201.
%e The a(3) = 86 set-systems:
%e {12}{13} {1}{2}{13}{123} {1}{2}{3}{13}{23}
%e {12}{23} {1}{2}{23}{123} {1}{2}{3}{13}{123}
%e {13}{23} {1}{3}{12}{13} {1}{2}{3}{23}{123}
%e {1}{2}{123} {1}{3}{12}{23} {1}{2}{12}{13}{23}
%e {1}{3}{123} {1}{3}{12}{123} {1}{2}{12}{13}{123}
%e {1}{12}{13} {1}{3}{13}{23} {1}{2}{12}{23}{123}
%e {1}{12}{23} {1}{3}{13}{123} {1}{2}{13}{23}{123}
%e {1}{12}{123} {1}{3}{23}{123} {1}{3}{12}{13}{23}
%e {1}{13}{23} {1}{12}{13}{23} {1}{3}{12}{13}{123}
%e {1}{13}{123} {1}{12}{13}{123} {1}{3}{12}{23}{123}
%e {2}{3}{123} {1}{12}{23}{123} {1}{3}{13}{23}{123}
%e {2}{12}{13} {1}{13}{23}{123} {1}{12}{13}{23}{123}
%e {2}{12}{23} {2}{3}{12}{13} {2}{3}{12}{13}{23}
%e {2}{12}{123} {2}{3}{12}{23} {2}{3}{12}{13}{123}
%e {2}{13}{23} {2}{3}{12}{123} {2}{3}{12}{23}{123}
%e {2}{23}{123} {2}{3}{13}{23} {2}{3}{13}{23}{123}
%e {3}{12}{13} {2}{3}{13}{123} {2}{12}{13}{23}{123}
%e {3}{12}{23} {2}{3}{23}{123} {3}{12}{13}{23}{123}
%e {3}{13}{23} {2}{12}{13}{23} {1}{2}{3}{12}{13}{23}
%e {3}{13}{123} {2}{12}{13}{123} {1}{2}{3}{12}{13}{123}
%e {3}{23}{123} {2}{12}{23}{123} {1}{2}{3}{12}{23}{123}
%e {12}{13}{23} {2}{13}{23}{123} {1}{2}{3}{13}{23}{123}
%e {12}{13}{123} {3}{12}{13}{23} {1}{2}{12}{13}{23}{123}
%e {12}{23}{123} {3}{12}{13}{123} {1}{3}{12}{13}{23}{123}
%e {13}{23}{123} {3}{12}{23}{123} {2}{3}{12}{13}{23}{123}
%e {1}{2}{3}{123} {3}{13}{23}{123} {1}{2}{3}{12}{13}{23}{123}
%e {1}{2}{12}{13} {12}{13}{23}{123}
%e {1}{2}{12}{23} {1}{2}{3}{12}{13}
%e {1}{2}{12}{123} {1}{2}{3}{12}{23}
%e {1}{2}{13}{23} {1}{2}{3}{12}{123}
%t dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
%t 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]]]]]]]]];
%t Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],Union@@#==Range[n]&&Length[csm[#]]<=1&&UnsameQ@@dual[#]&]],{n,0,3}]
%Y The same with covering instead of connected is A059201, with unlabeled version A319637.
%Y The non-T_0 version is A323818 (covering) or A326951 (not-covering).
%Y The non-connected version is A326940, with unlabeled version A326946.
%Y Cf. A000371, A003465, A245567, A316978, A319559, A319564, A326939, A326941, A326947.
%K nonn
%O 0,3
%A _Gus Wiseman_, Aug 08 2019
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