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A326948
Number of connected T_0 set-systems on n vertices.
4
1, 1, 3, 86, 31302, 2146841520, 9223371978880250448, 170141183460469231408869283342774399392, 57896044618658097711785492504343953919148780260559635830120038252613826101856
OFFSET
0,3
COMMENTS
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).
LINKS
FORMULA
Logarithmic transform of A059201.
EXAMPLE
The a(3) = 86 set-systems:
{12}{13} {1}{2}{13}{123} {1}{2}{3}{13}{23}
{12}{23} {1}{2}{23}{123} {1}{2}{3}{13}{123}
{13}{23} {1}{3}{12}{13} {1}{2}{3}{23}{123}
{1}{2}{123} {1}{3}{12}{23} {1}{2}{12}{13}{23}
{1}{3}{123} {1}{3}{12}{123} {1}{2}{12}{13}{123}
{1}{12}{13} {1}{3}{13}{23} {1}{2}{12}{23}{123}
{1}{12}{23} {1}{3}{13}{123} {1}{2}{13}{23}{123}
{1}{12}{123} {1}{3}{23}{123} {1}{3}{12}{13}{23}
{1}{13}{23} {1}{12}{13}{23} {1}{3}{12}{13}{123}
{1}{13}{123} {1}{12}{13}{123} {1}{3}{12}{23}{123}
{2}{3}{123} {1}{12}{23}{123} {1}{3}{13}{23}{123}
{2}{12}{13} {1}{13}{23}{123} {1}{12}{13}{23}{123}
{2}{12}{23} {2}{3}{12}{13} {2}{3}{12}{13}{23}
{2}{12}{123} {2}{3}{12}{23} {2}{3}{12}{13}{123}
{2}{13}{23} {2}{3}{12}{123} {2}{3}{12}{23}{123}
{2}{23}{123} {2}{3}{13}{23} {2}{3}{13}{23}{123}
{3}{12}{13} {2}{3}{13}{123} {2}{12}{13}{23}{123}
{3}{12}{23} {2}{3}{23}{123} {3}{12}{13}{23}{123}
{3}{13}{23} {2}{12}{13}{23} {1}{2}{3}{12}{13}{23}
{3}{13}{123} {2}{12}{13}{123} {1}{2}{3}{12}{13}{123}
{3}{23}{123} {2}{12}{23}{123} {1}{2}{3}{12}{23}{123}
{12}{13}{23} {2}{13}{23}{123} {1}{2}{3}{13}{23}{123}
{12}{13}{123} {3}{12}{13}{23} {1}{2}{12}{13}{23}{123}
{12}{23}{123} {3}{12}{13}{123} {1}{3}{12}{13}{23}{123}
{13}{23}{123} {3}{12}{23}{123} {2}{3}{12}{13}{23}{123}
{1}{2}{3}{123} {3}{13}{23}{123} {1}{2}{3}{12}{13}{23}{123}
{1}{2}{12}{13} {12}{13}{23}{123}
{1}{2}{12}{23} {1}{2}{3}{12}{13}
{1}{2}{12}{123} {1}{2}{3}{12}{23}
{1}{2}{13}{23} {1}{2}{3}{12}{123}
MATHEMATICA
dual[eds_]:=Table[First/@Position[eds, x], {x, Union@@eds}];
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]]]]]]]]];
Table[Length[Select[Subsets[Subsets[Range[n], {1, n}]], Union@@#==Range[n]&&Length[csm[#]]<=1&&UnsameQ@@dual[#]&]], {n, 0, 3}]
CROSSREFS
The same with covering instead of connected is A059201, with unlabeled version A319637.
The non-T_0 version is A323818 (covering) or A326951 (not-covering).
The non-connected version is A326940, with unlabeled version A326946.
Sequence in context: A185142 A279020 A302947 * A159053 A172494 A279131
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 08 2019
STATUS
approved