A334306 Number of distinct acyclic orientations of the edges of a three-dimensional n-sided prism with complete graphs as faces.

60, 501, 58848, 3296790, 248516640, 24173031960, 2940529011840, 436606222187520, 77604399434419200, 16251945275067163200, 3957141527033037235200, 1107716943231412920806400, 353062303151154587659468800, 127059236390700005739355008000

Offset: 3

Author

Matthew Scroggs, Apr 22 2020

Keywords

• nonn

Comments

Take the edge graph of an n-gonal prism and replace each of its 2-dimensional facets with a complete graph. The edges of this graph are then oriented so that no cycles are formed. a(n) is the number of different ways to do this with results that are not rotations of reflections of each other.

a(3) is the number of reference elements needed when using the finite element method for a 3-dimensional problem with hexahedral cells if the orientations of the mesh entities are derived from a low-to-high ordering of the vertex numbers.

Examples

			For n=3, the n-sided prism is a triangular prism. The faces of this are two triangles and three squares. Putting complete graphs on these faces gives the graph that consists of the edges of a triangular prism with diagonal edges added to the three square faces. a(3) is the number of acyclic orientations of this graph.
For n=4, the n-sided prism is a cube prism. The faces of this are six squares. Putting complete graphs on these faces gives the graph that consists of the edges of a cube with diagonal edges added to all six square faces (the "16-cell"). a(4) is the number of acyclic orientations of this graph.
		

Links

• Matthew Scroggs, Python code to calculate A334306
• Eric Weisstein's World of Mathematics, 16-Cell (the n=4 graph).

Crossrefs

Cf. A334304.

Extensions

a(7)-a(16) from Andrew Howroyd, Apr 23 2020