cp's OEIS Frontend

A367772 Number of sets of nonempty subsets of {1..n} satisfying a strict version of the axiom of choice in more than one way.

%I A367772 #12 Jul 26 2024 11:54:47
%S A367772 0,0,1,23,1105,154941,66072394,88945612865,396990456067403
%N A367772 Number of sets of nonempty subsets of {1..n} satisfying a strict version of the axiom of choice in more than one way.
%C A367772 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 A367772 Wikipedia, <a href="https://en.wikipedia.org/wiki/Axiom_of_choice">Axiom of choice</a>.
%F A367772 A367903(n) + A367904(n) + a(n) = A058891(n).
%e A367772 Non-isomorphic representatives of the a(3) = 23 set-systems:
%e A367772   {{1,2}}
%e A367772   {{1,2,3}}
%e A367772   {{1},{2,3}}
%e A367772   {{1},{1,2,3}}
%e A367772   {{1,2},{1,3}}
%e A367772   {{1,2},{1,2,3}}
%e A367772   {{1},{2,3},{1,2,3}}
%e A367772   {{1,2},{1,3},{2,3}}
%e A367772   {{1,2},{1,3},{1,2,3}}
%t A367772 Table[Length[Select[Subsets[Subsets[Range[n]]], Length[Select[Tuples[#], UnsameQ@@#&]]>1&]], {n,0,3}]
%Y A367772 For at least one choice we have A367902.
%Y A367772 For no choices we have A367903, no singletons A367769, ranks A367907.
%Y A367772 For a unique choice we have A367904, ranks A367908.
%Y A367772 These set-systems have ranks A367909.
%Y A367772 A000372 counts antichains, covering A006126, nonempty A014466.
%Y A367772 A003465 counts covering set-systems, unlabeled A055621.
%Y A367772 A058891 counts set-systems, unlabeled A000612.
%Y A367772 Cf. A059201, A102896, A133686, A283877, A306445, A323818, A355741, A367770, A367862, A367869, A367901, A367905.
%K A367772 nonn,more
%O A367772 0,4
%A A367772 _Gus Wiseman_, Dec 12 2023
%E A367772 a(5)-a(8) from _Christian Sievers_, Jul 26 2024