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A368984 Number of graphs with loops (symmetric relations) on n unlabeled vertices in which each connected component has an equal number of vertices and edges.

%I A368984 #12 Jan 29 2024 18:05:41
%S A368984 1,1,2,5,12,29,75,191,504,1339,3610,9800,26881,74118,205706,573514,
%T A368984 1606107,4513830,12727944,35989960,102026638,289877828,825273050,
%U A368984 2353794251,6724468631,19239746730,55123700591,158133959239,454168562921,1305796834570,3758088009136
%N A368984 Number of graphs with loops (symmetric relations) on n unlabeled vertices in which each connected component has an equal number of vertices and edges.
%C A368984 The graphs considered here can have loops but not parallel edges.
%C A368984 Also the number of unlabeled loop-graphs with n edges and n vertices such that it is possible to choose a different vertex from each edge. - _Gus Wiseman_, Jan 25 2024
%H A368984 Andrew Howroyd, <a href="/A368984/b368984.txt">Table of n, a(n) for n = 0..500</a>
%F A368984 Euler transform of A368983.
%e A368984 Representatives of the a(3) = 5 graphs are:
%e A368984    {{1,2}, {1,3}, {2,3}},
%e A368984    {{1}, {1,2}, {1,3}},
%e A368984    {{1}, {1,2}, {2,3}},
%e A368984    {{1}, {2}, {2,3}},
%e A368984    {{1}, {2}, {3}}.
%e A368984 The graph with 4 vertices and edges {{1}, {2}, {1,2}, {3,4}} is included by A368599 but not by this sequence.
%t A368984 brute[m_]:=First[Sort[Table[Sort[Sort/@(m/.Rule@@@Table[{(Union@@m)[[i]],p[[i]]},{i,Length[p]}])],{p,Permutations[Range[Length[Union@@m]]]}]]];
%t A368984 Table[Length[Union[brute/@Select[Subsets[Subsets[Range[n],{1,2}],{n}],Length[Select[Tuples[#],UnsameQ@@#&]]!=0&]]],{n,0,5}] (* _Gus Wiseman_, Jan 25 2024 *)
%Y A368984 The case of a unique choice is A000081.
%Y A368984 Without loops we have A137917, labeled A137916.
%Y A368984 The labeled version appears to be A333331.
%Y A368984 Without the choice condition we have A368598, covering A368599.
%Y A368984 The complement is counted by A368835, labeled A368596 (covering A368730).
%Y A368984 Row sums of A368926, labeled A368924.
%Y A368984 The connected case is A368983.
%Y A368984 A000085, A100861, A111924 count set partitions into singletons or pairs.
%Y A368984 A000666 counts unlabeled loop-graphs, covering A322700.
%Y A368984 A006125 counts simple graphs, unlabeled A000088.
%Y A368984 A006129 counts covering graphs, connected A001187, unlabeled A002494.
%Y A368984 A322661 counts labeled covering loop-graphs, connected A062740.
%Y A368984 Cf. A014068, A057500, A116508, A129271, A133686, A367863, A367869, A367902, A368597, A368601, A368836.
%K A368984 nonn
%O A368984 0,3
%A A368984 _Andrew Howroyd_, Jan 11 2024