This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A334304 #28 Feb 16 2025 08:34:00 %S A334304 1,1,3,501 %N A334304 Number of distinct acyclic orientations of the edges of an n-dimensional cube with complete graphs as facets. %C A334304 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. %C A334304 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. %H A334304 Matthew Scroggs, <a href="https://github.com/mscroggs/acyclic-orientations/blob/master/a334304.py">Python code to calculate A334304</a> %H A334304 Matthew W. Scroggs, Jørgen S. Dokken, Chris N. Richardson, Garth N. Wells, <a href="https://arxiv.org/abs/2102.11901">Construction of arbitrary order finite element degree-of-freedom maps on polygonal and polyhedral cell meshes</a>, arXiv:2102.11901 [math.NA], 2021. %H A334304 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/16-Cell.html">16-Cell</a> (the n=3 graph). %F A334304 A334248(n) <= a(n) <= A000142(2^n). %e A334304 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. %e A334304 There are 3 distinct (under rotations and reflections) acyclic orientations of the edges of this graph: %e A334304 *->-* *->-* *-<-* %e A334304 | | | | | | %e A334304 ^ ^ ^ v ^ v %e A334304 | | | | | | %e A334304 *->-* *->-* *->-* %e A334304 Therefore a(2) = 3. %e A334304 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. %Y A334304 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. %K A334304 nonn,more %O A334304 0,3 %A A334304 _Matthew Scroggs_, Apr 22 2020