A334304 Number of distinct acyclic orientations of the edges of an n-dimensional cube with complete graphs as facets.

1, 1, 3, 501

Offset: 0

Author

Matthew Scroggs, Apr 22 2020

Keywords

• nonn
• more

Comments

Take the edge graph of an n-dimensional cube and replace each of its (n-1) 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.

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

Examples

			For n=2, the n-dimensional cube is a square, and the (n-1)-dimensional facets are the edges of the square. Replacing the edges with complete graphs on 2 vertices does not change the graph.
There are 3 distinct (under rotations and reflections) acyclic orientations of the edges of this graph:
*->-*    *->-*    *-<-*
|   |    |   |    |   |
^   ^    ^   v    ^   v
|   |    |   |    |   |
*->-*    *->-*    *->-*
Therefore a(2) = 3.
For n=3, the n-dimensional cube is a cube, and the (n-1)-dimensional facets are the faces of the cube. Replacing the faces with complete graphs on 4 vertices gives the graph that is the edges of a cube with diagonal edges added to each face (the "16-cell"). a(3) is the number of distinct acyclic orientations of this graph.
		

Links

• Matthew Scroggs, Python code to calculate A334304
• Matthew W. Scroggs, Jørgen S. Dokken, Chris N. Richardson, Garth N. Wells, Construction of arbitrary order finite element degree-of-freedom maps on polygonal and polyhedral cell meshes, arXiv:2102.11901 [math.NA], 2021.
• Eric Weisstein's World of Mathematics, 16-Cell (the n=3 graph).

Crossrefs

A334248 is the number of distinct acyclic orientations of a n-cube (without the addition of complete graphs). A000012 is the number of reference elements needed when using the finite element method for an n-dimensional problem with simplectic cells.

Formula

A334248(n) <= a(n) <= A000142(2^n).