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A326219 Number of labeled n-vertex Hamiltonian digraphs (without loops).

%I A326219 #7 Jun 16 2019 22:33:56
%S A326219 0,1,1,15,1194
%N A326219 Number of labeled n-vertex Hamiltonian digraphs (without loops).
%C A326219 A digraph is Hamiltonian if it contains a directed cycle passing through every vertex exactly once.
%H A326219 Wikipedia, <a href="https://en.wikipedia.org/wiki/Hamiltonian_path">Hamiltonian path</a>
%F A326219 A053763(n) = a(n) + A326218(n).
%e A326219 The a(3) = 15 edge-sets:
%e A326219   {12,23,31}  {12,13,21,32}  {12,13,21,23,31}  {12,13,21,23,31,32}
%e A326219   {13,21,32}  {12,13,23,31}  {12,13,21,23,32}
%e A326219               {12,21,23,31}  {12,13,21,31,32}
%e A326219               {12,23,31,32}  {12,13,23,31,32}
%e A326219               {13,21,23,32}  {12,21,23,31,32}
%e A326219               {13,21,31,32}  {13,21,23,31,32}
%t A326219 Table[Length[Select[Subsets[Select[Tuples[Range[n],2],UnsameQ@@#&]],FindHamiltonianCycle[Graph[Range[n],DirectedEdge@@@#]]!={}&]],{n,0,4}] (* Mathematica 8.0+. Warning: Using HamiltonianGraphQ instead of FindHamiltonianCycle returns a(4) = 1200 which is incorrect *)
%Y A326219 The unlabeled case is A326225.
%Y A326219 The undirected case is A326208 (without loops) or A326240 (with loops).
%Y A326219 The case with loops is A326204.
%Y A326219 Digraphs (without loops) not containing a Hamiltonian cycle are A326218.
%Y A326219 Digraphs (without loops) containing a Hamiltonian path are A326217.
%Y A326219 Cf. A000595, A002416, A003024, A003216, A053763, A246446, A326220, A326226.
%K A326219 nonn,more
%O A326219 0,4
%A A326219 _Gus Wiseman_, Jun 15 2019