OFFSET
0,3
COMMENTS
A set-system is a finite set of finite nonempty sets. 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).
FORMULA
EXAMPLE
The a(0) = 1 through a(3) = 12 set-systems:
{} {{1}} {{1},{1,2}} {{1},{1,2},{1,3}}
{{2},{1,2}} {{2},{1,2},{2,3}}
{{3},{1,3},{2,3}}
{{1},{1,2},{1,2,3}}
{{1},{1,3},{1,2,3}}
{{2},{1,2},{1,2,3}}
{{2},{2,3},{1,2,3}}
{{3},{1,3},{1,2,3}}
{{3},{2,3},{1,2,3}}
{{1},{1,2},{1,3},{1,2,3}}
{{2},{1,2},{2,3},{1,2,3}}
{{3},{1,3},{2,3},{1,2,3}}
MATHEMATICA
dual[eds_]:=Table[First/@Position[eds, x], {x, Union@@eds}];
Table[Length[Select[Subsets[Subsets[Range[n], {1, n}]], Union@@#==Range[n]&&UnsameQ@@dual[#]&&SubsetQ[#, Intersection@@@Tuples[#, 2]]&]], {n, 0, 3}]
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Aug 11 2019
STATUS
approved