%I #15 Dec 29 2023 16:41:13

%S 1,1,1,2,20,308,5338,105298,2366704,60065072,1702900574,53400243419,

%T 1836274300504,68730359299960,2782263907231153,121137565273808792,

%U 5645321914669112342,280401845830658755142,14788386825536445299398,825378055206721558026931,48604149005046792753887416

%N Number of n-vertex labeled simple graphs with the same number of edges as covered vertices.

%C Unlike the connected case (A057500), these graphs may have more than one cycle; for example, the graph {{1,2},{1,3},{1,4},{2,3},{2,4},{5,6}} has multiple cycles.

%H Andrew Howroyd, <a href="/A367862/b367862.txt">Table of n, a(n) for n = 0..200</a>

%F Binomial transform of A367863.

%e Non-isomorphic representatives of the a(4) = 20 graphs:

%e {}

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

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

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

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

%o (PARI) \\ Here b(n) is A367863(n)

%o b(n) = sum(k=0, n, (-1)^(n-k) * binomial(n,k) * binomial(binomial(k,2), n))

%o a(n) = sum(k=0, n, binomial(n,k) * b(k)) \\ _Andrew Howroyd_, Dec 29 2023

%Y The connected case is A057500, unlabeled A001429.

%Y Counting all vertices (not just covered) gives A116508.

%Y The covering case is A367863, unlabeled A006649.

%Y For set-systems we have A367916, ranks A367917.

%Y A001187 counts connected graphs, A001349 unlabeled.

%Y A003465 counts covering set-systems, unlabeled A055621, ranks A326754.

%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 A133686 = graphs satisfy strict AoC, connected A129271, covering A367869.

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

%Y A323818 counts connected set-systems, unlabeled A323819, ranks A326749.

%Y A367867 = graphs contradict strict AoC, connected A140638, covering A367868.

%Y Cf. A006126, A316983, A321405, A326754, A326870, A326877, A367770, A367901, A367902, A367903.

%K nonn

%O 0,4

%A _Gus Wiseman_, Dec 07 2023

%E Terms a(8) and beyond from _Andrew Howroyd_, Dec 29 2023