A326204 Number of Hamiltonian labeled n-vertex digraphs (with loops).

0, 2, 4, 120, 19104, 13531200, 38124171520

Offset

0,2

Comments

A digraph is Hamiltonian if it contains a directed cycle passing through every vertex exactly once.

Links

Table of n, a(n) for n=0..6.

Wikipedia, Hamiltonian path.

Formula

a(n) = A002416(n)-A326220(n). - Pontus von Brömssen, Jan 06 2026

Example

The a(2) = 4 digraph edge-sets:
  {12,21}
  {11,12,21}
  {12,21,22}
  {11,12,21,22}

Mathematica

Table[Length[Select[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) = 19200 which is incorrect *)

Crossrefs

The unlabeled case is A326226.

The case without loops is A326219.

The undirected case (without loops) is A326208.

Non-Hamiltonian digraphs are A326220.

Digraphs containing a Hamiltonian path are A326214.

Cf. A000595, A002416, A003024, A003216, A057864.

Sequence in context: A132497 A009484 A006314 * A259381 A333621 A009595

Adjacent sequences: A326201 A326202 A326203 * A326205 A326206 A326207

Keyword

nonn,more

Author

Gus Wiseman, Jun 14 2019

Extensions

a(5)-a(6) (using A326220) from Pontus von Brömssen, Jan 06 2026

Status

approved