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%I A367903 #20 Jul 30 2024 14:41:26
%S A367903 0,0,1,67,30997,2147296425,9223372036784737528,
%T A367903 170141183460469231731687303625772608225,
%U A367903 57896044618658097711785492504343953926634992332820282019728791606173188627779
%N A367903 Number of sets of nonempty subsets of {1..n} contradicting a strict version of the axiom of choice.
%C A367903 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 A367903 Wikipedia, <a href="https://en.wikipedia.org/wiki/Axiom_of_choice">Axiom of choice</a>.
%F A367903 a(n) + A367904(n) + A367772(n) = A058891(n+1) = 2^(2^n-1).
%e A367903 The a(2) = 1 set-system is {{1},{2},{1,2}}.
%e A367903 The a(3) = 67 set-systems have following 21 non-isomorphic representatives:
%e A367903   {{1},{2},{1,2}}
%e A367903   {{1},{2},{3},{1,2}}
%e A367903   {{1},{2},{3},{1,2,3}}
%e A367903   {{1},{2},{1,2},{1,3}}
%e A367903   {{1},{2},{1,2},{1,2,3}}
%e A367903   {{1},{2},{1,3},{2,3}}
%e A367903   {{1},{2},{1,3},{1,2,3}}
%e A367903   {{1},{1,2},{1,3},{2,3}}
%e A367903   {{1},{1,2},{1,3},{1,2,3}}
%e A367903   {{1},{1,2},{2,3},{1,2,3}}
%e A367903   {{1,2},{1,3},{2,3},{1,2,3}}
%e A367903   {{1},{2},{3},{1,2},{1,3}}
%e A367903   {{1},{2},{3},{1,2},{1,2,3}}
%e A367903   {{1},{2},{1,2},{1,3},{2,3}}
%e A367903   {{1},{2},{1,2},{1,3},{1,2,3}}
%e A367903   {{1},{2},{1,3},{2,3},{1,2,3}}
%e A367903   {{1},{1,2},{1,3},{2,3},{1,2,3}}
%e A367903   {{1},{2},{3},{1,2},{1,3},{2,3}}
%e A367903   {{1},{2},{3},{1,2},{1,3},{1,2,3}}
%e A367903   {{1},{2},{1,2},{1,3},{2,3},{1,2,3}}
%e A367903   {{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}
%t A367903 Table[Length[Select[Subsets[Rest[Subsets[Range[n]]]], Select[Tuples[#],UnsameQ@@#&]=={}&]],{n,0,3}]
%Y A367903 Multisets of multisets of this type are ranked by A355529.
%Y A367903 The version without singletons is A367769.
%Y A367903 The version for simple graphs is A367867, covering A367868.
%Y A367903 The version allowing empty edges is A367901.
%Y A367903 The complement is A367902, without singletons A367770, ranks A367906.
%Y A367903 For a unique choice (instead of none) we have A367904, ranks A367908.
%Y A367903 These set-systems have ranks A367907.
%Y A367903 An unlabeled version is A368094, for multiset partitions A368097.
%Y A367903 A000372 counts antichains, covering A006126, nonempty A014466.
%Y A367903 A003465 counts covering set-systems, unlabeled A055621.
%Y A367903 A058891 counts set-systems, unlabeled A000612.
%Y A367903 A059201 counts covering T_0 set-systems.
%Y A367903 A323818 counts covering connected set-systems.
%Y A367903 A326031 gives weight of the set-system with BII-number n.
%Y A367903 Cf. A007716, A083323, A092918, A102896, A283877, A306445, A355739, A355740, A367862, A367905, A368409, A368413.
%K A367903 nonn,more
%O A367903 0,4
%A A367903 _Gus Wiseman_, Dec 05 2023
%E A367903 a(5)-a(8) from _Christian Sievers_, Jul 26 2024

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