A367770 - OEIS

%I #12 Jul 28 2024 12:32:20

%S 1,1,2,15,558,81282,39400122,61313343278,309674769204452

%N Number of sets of nonempty non-singleton subsets of {1..n} satisfying 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 Excludes 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>.

%e The a(3) = 15 set-systems:

%e   {}

%e   {{1,2}}

%e   {{1,3}}

%e   {{2,3}}

%e   {{1,2,3}}

%e   {{1,2},{1,3}}

%e   {{1,2},{2,3}}

%e   {{1,2},{1,2,3}}

%e   {{1,3},{2,3}}

%e   {{1,3},{1,2,3}}

%e   {{2,3},{1,2,3}}

%e   {{1,2},{1,3},{2,3}}

%e   {{1,2},{1,3},{1,2,3}}

%e   {{1,2},{2,3},{1,2,3}}

%e   {{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 version for simple graphs is A133686, covering A367869.

%Y The complement is counted by A367769.

%Y The complement allowing singletons and empty sets is A367901.

%Y Allowing singletons gives A367902, ranks A367906.

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

%Y These set-systems have ranks A367906 /\ A326781.

%Y A000372 counts antichains, covering A006126, nonempty A014466.

%Y A003465 counts covering set-systems, unlabeled A055621.

%Y A058891 counts set-systems, unlabeled A000612.

%Y A323818 counts covering connected set-systems, unlabeled A323819.

%Y Cf. A059201, A083323, A092918, A102896, A283877, A305000, A306445, A355739, A355740, A367904, A367905.

%K nonn,more

%O 0,3

%A _Gus Wiseman_, Dec 05 2023

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