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%I #11 Dec 28 2023 09:22:15
%S 0,0,1,67,30997
%N Number of sets of nonempty subsets of {1..n} contradicting a strict version of the axiom of choice.
%C 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.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Axiom_of_choice">Axiom of choice</a>.
%F A367903(n) + A367904(n) + A367772(n) = A058891(n).
%e The a(2) = 1 set-system is {{1},{2},{1,2}}.
%e The a(3) = 67 set-systems have following 21 non-isomorphic representatives:
%e {{1},{2},{1,2}}
%e {{1},{2},{3},{1,2}}
%e {{1},{2},{3},{1,2,3}}
%e {{1},{2},{1,2},{1,3}}
%e {{1},{2},{1,2},{1,2,3}}
%e {{1},{2},{1,3},{2,3}}
%e {{1},{2},{1,3},{1,2,3}}
%e {{1},{1,2},{1,3},{2,3}}
%e {{1},{1,2},{1,3},{1,2,3}}
%e {{1},{1,2},{2,3},{1,2,3}}
%e {{1,2},{1,3},{2,3},{1,2,3}}
%e {{1},{2},{3},{1,2},{1,3}}
%e {{1},{2},{3},{1,2},{1,2,3}}
%e {{1},{2},{1,2},{1,3},{2,3}}
%e {{1},{2},{1,2},{1,3},{1,2,3}}
%e {{1},{2},{1,3},{2,3},{1,2,3}}
%e {{1},{1,2},{1,3},{2,3},{1,2,3}}
%e {{1},{2},{3},{1,2},{1,3},{2,3}}
%e {{1},{2},{3},{1,2},{1,3},{1,2,3}}
%e {{1},{2},{1,2},{1,3},{2,3},{1,2,3}}
%e {{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}
%t Table[Length[Select[Subsets[Rest[Subsets[Range[n]]]], Select[Tuples[#],UnsameQ@@#&]=={}&]],{n,0,3}]
%Y Multisets of multisets of this type are ranked by A355529.
%Y The version without singletons is A367769.
%Y The version for simple graphs is A367867, covering A367868.
%Y The version allowing empty edges is A367901.
%Y The complement is A367902, without singletons A367770, ranks A367906.
%Y For a unique choice (instead of none) we have A367904, ranks A367908.
%Y These set-systems have ranks A367907.
%Y An unlabeled version is A368094, for multiset partitions A368097.
%Y A000372 counts antichains, covering A006126, nonempty A014466.
%Y A003465 counts covering set-systems, unlabeled A055621.
%Y A058891 counts set-systems, unlabeled A000612.
%Y A059201 counts covering T_0 set-systems.
%Y A323818 counts covering connected set-systems.
%Y A326031 gives weight of the set-system with BII-number n.
%Y Cf. A007716, A083323, A092918, A102896, A283877, A306445, A355739, A355740, A367862, A367905, A368409, A368413.
%K nonn,more
%O 0,4
%A _Gus Wiseman_, Dec 05 2023
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