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A326226 Number of unlabeled n-vertex Hamiltonian digraphs (with loops).

%I A326226 #14 Jun 17 2019 01:36:36
%S A326226 0,2,3,24,858
%N A326226 Number of unlabeled n-vertex Hamiltonian digraphs (with loops).
%C A326226 A digraph is Hamiltonian if it contains a directed cycle passing through every vertex exactly once.
%H A326226 Wikipedia, <a href="https://en.wikipedia.org/wiki/Hamiltonian_path">Hamiltonian path</a>
%H A326226 Gus Wiseman, <a href="/A326226/a326226.png">Non-isomorphic representatives of the a(3) = 24 Hamiltonian digraphs</a>
%e A326226 Non-isomorphic representatives of the a(2) = 3 digraph edge-sets:
%e A326226   {12,21}
%e A326226   {11,12,21}
%e A326226   {11,12,21,22}
%t A326226 dinorm[m_]:=If[m=={},{},If[Union@@m!=Range[Max@@Flatten[m]],dinorm[m/. Apply[Rule,Table[{(Union@@m)[[i]],i},{i,Length[Union@@m]}],{1}]],First[Sort[dinorm[m,1]]]]];
%t A326226 dinorm[m_,aft_]:=If[Length[Union@@m]<=aft,{m},With[{mx=Table[Count[m,i,{2}],{i,Select[Union@@m,#1>=aft&]}]},Union@@(dinorm[#1,aft+1]&)/@Union[Table[Map[Sort,m/. {par+aft-1->aft,aft->par+aft-1},{0}],{par,First/@Position[mx,Max[mx]]}]]]];
%t A326226 Table[Length[Select[Union[dinorm/@Subsets[Tuples[Range[n],2]]],FindHamiltonianCycle[Graph[Range[n],DirectedEdge@@@#]]!={}&]],{n,0,4}] (* Mathematica 8.0+. Warning: Using HamiltonianGraphQ instead of FindHamiltonianCycle returns a(4) = 867 which is incorrect *)
%Y A326226 The labeled case is A326204.
%Y A326226 The case without loops is A326225.
%Y A326226 The undirected case is A003216 (without loops) or A326215 (with loops).
%Y A326226 Unlabeled non-Hamiltonian digraphs are A326223.
%Y A326226 Unlabeled digraphs with a Hamiltonian path are A326221.
%Y A326226 Cf. A000595, A002416, A003087, A057864, A246446, A326208.
%K A326226 nonn,hard,more
%O A326226 0,2
%A A326226 _Gus Wiseman_, Jun 14 2019