cp's OEIS Frontend

A367770 Number of sets of nonempty non-singleton subsets of {1..n} satisfying a strict version of the axiom of choice.

%I A367770 #12 Jul 28 2024 12:32:20
%S A367770 1,1,2,15,558,81282,39400122,61313343278,309674769204452
%N A367770 Number of sets of nonempty non-singleton subsets of {1..n} satisfying a strict version of the axiom of choice.
%C A367770 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 A367770 Excludes all set-systems with more edges than covered vertices, but this condition is not sufficient.
%H A367770 Wikipedia, <a href="https://en.wikipedia.org/wiki/Axiom_of_choice">Axiom of choice</a>.
%e A367770 The a(3) = 15 set-systems:
%e A367770   {}
%e A367770   {{1,2}}
%e A367770   {{1,3}}
%e A367770   {{2,3}}
%e A367770   {{1,2,3}}
%e A367770   {{1,2},{1,3}}
%e A367770   {{1,2},{2,3}}
%e A367770   {{1,2},{1,2,3}}
%e A367770   {{1,3},{2,3}}
%e A367770   {{1,3},{1,2,3}}
%e A367770   {{2,3},{1,2,3}}
%e A367770   {{1,2},{1,3},{2,3}}
%e A367770   {{1,2},{1,3},{1,2,3}}
%e A367770   {{1,2},{2,3},{1,2,3}}
%e A367770   {{1,3},{2,3},{1,2,3}}
%t A367770 Table[Length[Select[Subsets[Select[Subsets[Range[n]], Length[#]>1&]], Select[Tuples[#], UnsameQ@@#&]!={}&]],{n,0,3}]
%Y A367770 Set-systems without singletons are counted by A016031, covering A323816.
%Y A367770 The version for simple graphs is A133686, covering A367869.
%Y A367770 The complement is counted by A367769.
%Y A367770 The complement allowing singletons and empty sets is A367901.
%Y A367770 Allowing singletons gives A367902, ranks A367906.
%Y A367770 The complement allowing singletons is A367903, ranks A367907.
%Y A367770 These set-systems have ranks A367906 /\ A326781.
%Y A367770 A000372 counts antichains, covering A006126, nonempty A014466.
%Y A367770 A003465 counts covering set-systems, unlabeled A055621.
%Y A367770 A058891 counts set-systems, unlabeled A000612.
%Y A367770 A323818 counts covering connected set-systems, unlabeled A323819.
%Y A367770 Cf. A059201, A083323, A092918, A102896, A283877, A305000, A306445, A355739, A355740, A367904, A367905.
%K A367770 nonn,more
%O A367770 0,3
%A A367770 _Gus Wiseman_, Dec 05 2023
%E A367770 a(6)-a(8) from _Christian Sievers_, Jul 28 2024