A367902 - OEIS

%I #11 Jul 25 2024 18:29:01

%S 1,2,7,61,1771,187223,70038280,90111497503,397783376192189

%N Number of sets of nonempty 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.

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

%F a(n) = A370636(2^n-1). - _Alois P. Heinz_, Mar 09 2024

%e The a(2) = 7 set-systems:

%e   {}

%e   {{1}}

%e   {{2}}

%e   {{1,2}}

%e   {{1},{2}}

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

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

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

%Y The version for simple graphs is A133686, covering A367869.

%Y The version without singletons is A367770.

%Y The complement allowing empty edges is A367901.

%Y The complement is A367903, without singletons A367769, ranks A367907.

%Y For a unique choice we have A367904, ranks A367908.

%Y These set-systems have ranks A367906.

%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, A370636.

%K nonn,more

%O 0,2

%A _Gus Wiseman_, Dec 05 2023

%E a(6)-a(8) from _Christian Sievers_, Jul 25 2024