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A326214 Number of labeled n-vertex digraphs (with loops) containing a (directed) Hamiltonian path.

%I A326214 #9 Aug 23 2023 08:36:29
%S A326214 0,0,12,384,53184
%N A326214 Number of labeled n-vertex digraphs (with loops) containing a (directed) Hamiltonian path.
%C A326214 A path is Hamiltonian if it passes through every vertex exactly once.
%H A326214 Wikipedia, <a href="https://en.wikipedia.org/wiki/Hamiltonian_path">Hamiltonian path</a>
%H A326214 Gus Wiseman, <a href="http://arxiv.org/abs/0709.0430">Enumeration of paths and cycles and e-coefficients of incomparability graphs</a>, arXiv:0709.0430 [math.CO], 2007.
%F A326214 A002416(n) = a(n) + A326213(n).
%e A326214 The a(2) = 12 edge-sets:
%e A326214   {12}
%e A326214   {21}
%e A326214   {11,12}
%e A326214   {11,21}
%e A326214   {12,21}
%e A326214   {12,22}
%e A326214   {21,22}
%e A326214   {11,12,21}
%e A326214   {11,12,22}
%e A326214   {11,21,22}
%e A326214   {12,21,22}
%e A326214   {11,12,21,22}
%t A326214 Table[Length[Select[Subsets[Tuples[Range[n],2]],FindHamiltonianPath[Graph[Range[n],DirectedEdge@@@#]]!={}&]],{n,4}] (* Mathematica 10.2+ *)
%Y A326214 The unlabeled case is A326221.
%Y A326214 The undirected case is A326206.
%Y A326214 The case without loops is A326217.
%Y A326214 Digraphs not containing a Hamiltonian path are A326213.
%Y A326214 Digraphs containing a Hamiltonian cycle are A326204.
%Y A326214 Cf. A000595, A002416, A003024, A003216, A057864, A326224, A326225, A326226.
%K A326214 nonn,more
%O A326214 0,3
%A A326214 _Gus Wiseman_, Jun 15 2019