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A326217
Number of labeled n-vertex digraphs (without loops) containing a Hamiltonian path.
7
0, 0, 3, 48, 3324, 929005, 1014750550, 4305572108670
OFFSET
0,3
FORMULA
A053763(n) = a(n) + A326216(n).
EXAMPLE
The a(3) = 48 edge-sets:
{12,23} {12,13,21} {12,13,21,23} {12,13,21,23,31} {12,13,21,23,31,32}
{12,31} {12,13,23} {12,13,21,31} {12,13,21,23,32}
{13,21} {12,13,31} {12,13,21,32} {12,13,21,31,32}
{13,32} {12,13,32} {12,13,23,31} {12,13,23,31,32}
{21,32} {12,21,23} {12,13,23,32} {12,21,23,31,32}
{23,31} {12,21,31} {12,13,31,32} {13,21,23,31,32}
{12,21,32} {12,21,23,31}
{12,23,31} {12,21,23,32}
{12,23,32} {12,21,31,32}
{12,31,32} {12,23,31,32}
{13,21,23} {13,21,23,31}
{13,21,31} {13,21,23,32}
{13,21,32} {13,21,31,32}
{13,23,31} {13,23,31,32}
{13,23,32} {21,23,31,32}
{13,31,32}
{21,23,31}
{21,23,32}
{21,31,32}
{23,31,32}
MATHEMATICA
Table[Length[Select[Subsets[Select[Tuples[Range[n], 2], UnsameQ@@#&]], FindHamiltonianPath[Graph[Range[n], DirectedEdge@@@#]]!={}&]], {n, 4}] (* Mathematica 10.2+ *)
CROSSREFS
The undirected case is A326206.
The unlabeled undirected case is A057864.
The case with loops is A326214.
Unlabeled digraphs with a Hamiltonian path are A326221.
Digraphs (without loops) not containing a Hamiltonian path are A326216.
Digraphs (without loops) containing a Hamiltonian cycle are A326219.
Sequence in context: A201698 A356612 A295813 * A003029 A049524 A270748
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Jun 15 2019
EXTENSIONS
a(5)-a(7) from Bert Dobbelaere, Feb 21 2023
STATUS
approved