%I #12 Dec 30 2023 17:40:21

%S 0,0,0,0,7,381,21853,1790135,250562543,66331467215,34507857686001,

%T 35645472109753873,73356936892660012513,301275024409580265134121,

%U 2471655539736293803311467943,40527712706903494712385171632959,1328579255614092966328511889576785109

%N Number of labeled simple graphs covering n vertices and 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.

%H Andrew Howroyd, <a href="/A367868/b367868.txt">Table of n, a(n) for n = 0..50</a>

%F a(n) = A006129(n) - A367869(n). - _Andrew Howroyd_, Dec 30 2023

%e The a(4) = 7 graphs:

%e {{1,2},{1,3},{1,4},{2,3},{2,4}}

%e {{1,2},{1,3},{1,4},{2,3},{3,4}}

%e {{1,2},{1,3},{1,4},{2,4},{3,4}}

%e {{1,2},{1,3},{2,3},{2,4},{3,4}}

%e {{1,2},{1,4},{2,3},{2,4},{3,4}}

%e {{1,3},{1,4},{2,3},{2,4},{3,4}}

%e {{1,2},{1,3},{1,4},{2,3},{2,4},{3,4}}

%t Table[Length[Select[Subsets[Subsets[Range[n],{2}]], Union@@#==Range[n]&&Select[Tuples[#], UnsameQ@@#&]=={}&]],{n,0,5}]

%Y The connected case is A140638, unlabeled A140636.

%Y The non-covering case is A367867.

%Y The complement is A367869, connected A129271, non-covering A133686.

%Y The version for set-systems is A367903, ranks A367907.

%Y A001187 counts connected graphs, A001349 unlabeled.

%Y A006125 counts graphs, A000088 unlabeled.

%Y A006129 counts covering graphs, A002494 unlabeled.

%Y A058891 counts set-systems (without singletons A016031), unlabeled A000612.

%Y A059201 counts covering T_0 set-systems, unlabeled A319637, ranks A326947.

%Y A143543 counts simple labeled graphs by number of connected components.

%Y Cf. A057500, A116508, A367769, A367770, A367863, A367901, A367902, A367904.

%K nonn

%O 0,5

%A _Gus Wiseman_, Dec 08 2023

%E Terms a(7) and beyond from _Andrew Howroyd_, Dec 30 2023