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Number of finite sets of nonempty non-singleton subsets of {1..n} contradicting a strict version of the axiom of choice.
20

%I #16 Jul 28 2024 12:30:57

%S 0,0,0,1,1490,67027582,144115188036455750,

%T 1329227995784915872903806998967001298,

%U 226156424291633194186662080095093570025917938800079226639565284090686126876

%N Number of finite sets of nonempty non-singleton 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.

%C Includes all set-systems with more edges than covered vertices, but this condition is not sufficient.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Axiom_of_choice">Axiom of choice</a>.

%F a(n) = 2^(2^n-n-1) - A367770(n) = A016031(n+1) - A367770(n). - _Christian Sievers_, Jul 28 2024

%e The a(3) = 1 set-system is: {{1,2},{1,3},{2,3},{1,2,3}}.

%t Table[Length[Select[Subsets[Select[Subsets[Range[n]], Length[#]>1&]], Select[Tuples[#], UnsameQ@@#&]=={}&]], {n,0,3}]

%Y Set-systems without singletons are counted by A016031, covering A323816.

%Y The complement is A367770, with singletons allowed A367902 (ranks A367906).

%Y The version for simple graphs is A367867, covering A367868.

%Y The version allowing singletons and empty edges is A367901.

%Y The version allowing singletons is A367903, ranks A367907.

%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 Cf. A007716, A092918, A102896, A283877, A306445, A326031, A355739, A355740, A355741, A367904, A367905.

%K nonn,more

%O 0,5

%A _Gus Wiseman_, Dec 05 2023

%E a(6)-a(8) from _Christian Sievers_, Jul 28 2024