cp's OEIS Frontend

A367862 Number of n-vertex labeled simple graphs with the same number of edges as covered vertices.

%I A367862 #15 Dec 29 2023 16:41:13
%S A367862 1,1,1,2,20,308,5338,105298,2366704,60065072,1702900574,53400243419,
%T A367862 1836274300504,68730359299960,2782263907231153,121137565273808792,
%U A367862 5645321914669112342,280401845830658755142,14788386825536445299398,825378055206721558026931,48604149005046792753887416
%N A367862 Number of n-vertex labeled simple graphs with the same number of edges as covered vertices.
%C A367862 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 A367862 Andrew Howroyd, <a href="/A367862/b367862.txt">Table of n, a(n) for n = 0..200</a>
%F A367862 Binomial transform of A367863.
%e A367862 Non-isomorphic representatives of the a(4) = 20 graphs:
%e A367862   {}
%e A367862   {{1,2},{1,3},{2,3}}
%e A367862   {{1,2},{1,3},{1,4},{2,3}}
%e A367862   {{1,2},{1,3},{2,4},{3,4}}
%t A367862 Table[Length[Select[Subsets[Subsets[Range[n],{2}]], Length[#]==Length[Union@@#]&]],{n,0,5}]
%o A367862 (PARI) \\ Here b(n) is A367863(n)
%o A367862 b(n) = sum(k=0, n, (-1)^(n-k) * binomial(n,k) * binomial(binomial(k,2), n))
%o A367862 a(n) = sum(k=0, n, binomial(n,k) * b(k)) \\ _Andrew Howroyd_, Dec 29 2023
%Y A367862 The connected case is A057500, unlabeled A001429.
%Y A367862 Counting all vertices (not just covered) gives A116508.
%Y A367862 The covering case is A367863, unlabeled A006649.
%Y A367862 For set-systems we have A367916, ranks A367917.
%Y A367862 A001187 counts connected graphs, A001349 unlabeled.
%Y A367862 A003465 counts covering set-systems, unlabeled A055621, ranks A326754.
%Y A367862 A006125 counts graphs, A000088 unlabeled.
%Y A367862 A006129 counts covering graphs, A002494 unlabeled.
%Y A367862 A058891 counts set-systems, unlabeled A000612, without singletons A016031.
%Y A367862 A059201 counts covering T_0 set-systems, unlabeled A319637, ranks A326947.
%Y A367862 A133686 = graphs satisfy strict AoC, connected A129271, covering A367869.
%Y A367862 A143543 counts simple labeled graphs by number of connected components.
%Y A367862 A323818 counts connected set-systems, unlabeled A323819, ranks A326749.
%Y A367862 A367867 = graphs contradict strict AoC, connected A140638, covering A367868.
%Y A367862 Cf. A006126, A316983, A321405, A326754, A326870, A326877, A367770, A367901, A367902, A367903.
%K A367862 nonn
%O A367862 0,4
%A A367862 _Gus Wiseman_, Dec 07 2023
%E A367862 Terms a(8) and beyond from _Andrew Howroyd_, Dec 29 2023