A367869 - OEIS

%I #11 Dec 30 2023 17:38:56

%S 1,0,1,4,34,387,5596,97149,1959938,44956945,1154208544,32772977715,

%T 1019467710328,34473686833527,1259038828370402,49388615245426933,

%U 2070991708598960524,92445181295983865757,4376733266230674345874,219058079619119072854095,11556990682657196214302036

%N Number of labeled simple graphs covering n vertices and 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 Number of labeled n-node graphs with at most one cycle in each component and no isolated vertices. - _Andrew Howroyd_, Dec 30 2023

%H Andrew Howroyd, <a href="/A367869/b367869.txt">Table of n, a(n) for n = 0..200</a>

%F E.g.f.: exp(B(x) - x - 1) where B(x) is the e.g.f. of A129271. - _Andrew Howroyd_, Dec 30 2023

%e The a(3) = 4 graphs:

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

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

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

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

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

%o (PARI) seq(n)={my(t=-lambertw(-x + O(x*x^n))); Vec(serlaplace(sqrt(1/(1-t))*exp(t/2 - 3*t^2/4 - x)))} \\ _Andrew Howroyd_, Dec 30 2023

%Y The connected case is A129271.

%Y The non-covering case is A133686, complement A367867.

%Y The complement is A367868, connected A140638 (unlabeled A140636).

%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, unlabeled A000612, without singletons A016031.

%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, A355740, A367770, A367863, A367902.

%K nonn

%O 0,4

%A _Gus Wiseman_, Dec 08 2023

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