OFFSET
0,4
COMMENTS
The axiom of choice says that, given any set of nonempty sets Y, it is possible to choose a set containing an element from each. The strict version requires this set to have the same cardinality as Y, meaning no element is chosen more than once.
LINKS
Wikipedia, Axiom of choice.
EXAMPLE
The a(2) = 1 set-system is {{1},{2},{1,2}}.
The a(3) = 67 set-systems have following 21 non-isomorphic representatives:
{{1},{2},{1,2}}
{{1},{2},{3},{1,2}}
{{1},{2},{3},{1,2,3}}
{{1},{2},{1,2},{1,3}}
{{1},{2},{1,2},{1,2,3}}
{{1},{2},{1,3},{2,3}}
{{1},{2},{1,3},{1,2,3}}
{{1},{1,2},{1,3},{2,3}}
{{1},{1,2},{1,3},{1,2,3}}
{{1},{1,2},{2,3},{1,2,3}}
{{1,2},{1,3},{2,3},{1,2,3}}
{{1},{2},{3},{1,2},{1,3}}
{{1},{2},{3},{1,2},{1,2,3}}
{{1},{2},{1,2},{1,3},{2,3}}
{{1},{2},{1,2},{1,3},{1,2,3}}
{{1},{2},{1,3},{2,3},{1,2,3}}
{{1},{1,2},{1,3},{2,3},{1,2,3}}
{{1},{2},{3},{1,2},{1,3},{2,3}}
{{1},{2},{3},{1,2},{1,3},{1,2,3}}
{{1},{2},{1,2},{1,3},{2,3},{1,2,3}}
{{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}
MATHEMATICA
Table[Length[Select[Subsets[Rest[Subsets[Range[n]]]], Select[Tuples[#], UnsameQ@@#&]=={}&]], {n, 0, 3}]
CROSSREFS
Multisets of multisets of this type are ranked by A355529.
The version without singletons is A367769.
The version allowing empty edges is A367901.
These set-systems have ranks A367907.
A059201 counts covering T_0 set-systems.
A323818 counts covering connected set-systems.
A326031 gives weight of the set-system with BII-number n.
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Dec 05 2023
EXTENSIONS
a(5)-a(8) from Christian Sievers, Jul 26 2024
STATUS
approved