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%I #12 Aug 23 2023 08:36:41
%S 0,0,3,48,3324,929005,1014750550,4305572108670
%N Number of labeled n-vertex digraphs (without loops) containing a Hamiltonian path.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Hamiltonian_path">Hamiltonian path</a>
%H 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 A053763(n) = a(n) + A326216(n).
%e The a(3) = 48 edge-sets:
%e {12,23} {12,13,21} {12,13,21,23} {12,13,21,23,31} {12,13,21,23,31,32}
%e {12,31} {12,13,23} {12,13,21,31} {12,13,21,23,32}
%e {13,21} {12,13,31} {12,13,21,32} {12,13,21,31,32}
%e {13,32} {12,13,32} {12,13,23,31} {12,13,23,31,32}
%e {21,32} {12,21,23} {12,13,23,32} {12,21,23,31,32}
%e {23,31} {12,21,31} {12,13,31,32} {13,21,23,31,32}
%e {12,21,32} {12,21,23,31}
%e {12,23,31} {12,21,23,32}
%e {12,23,32} {12,21,31,32}
%e {12,31,32} {12,23,31,32}
%e {13,21,23} {13,21,23,31}
%e {13,21,31} {13,21,23,32}
%e {13,21,32} {13,21,31,32}
%e {13,23,31} {13,23,31,32}
%e {13,23,32} {21,23,31,32}
%e {13,31,32}
%e {21,23,31}
%e {21,23,32}
%e {21,31,32}
%e {23,31,32}
%t Table[Length[Select[Subsets[Select[Tuples[Range[n],2],UnsameQ@@#&]],FindHamiltonianPath[Graph[Range[n],DirectedEdge@@@#]]!={}&]],{n,4}] (* Mathematica 10.2+ *)
%Y The undirected case is A326206.
%Y The unlabeled undirected case is A057864.
%Y The case with loops is A326214.
%Y Unlabeled digraphs with a Hamiltonian path are A326221.
%Y Digraphs (without loops) not containing a Hamiltonian path are A326216.
%Y Digraphs (without loops) containing a Hamiltonian cycle are A326219.
%Y Cf. A000595, A002416, A003024, A003216, A283420, A326204, A326208, A326213, A326225.
%K nonn,more
%O 0,3
%A _Gus Wiseman_, Jun 15 2019
%E a(5)-a(7) from _Bert Dobbelaere_, Feb 21 2023