I have to design a combinatorial algorithm based on some simmetries of an hypercube and I'm pretty sure such a problem has already been studied. Let's start with a 3D case. Consider a cube like the one below, i can divide it into tetrahedra. I can divide the cube with 6 tetrahedra. The observation I need to formalize how many tetrahedra share a given edge/face etc of a given tetrahedron. Looking at my original cube, lets consider the tetrahedron, DHFE, the edge DF would be shared by all the others tetrahedron, the edge DE and HF would be shared by two tetrahedron at most and so on, similar observation can be done for the face DEF as example. Let's say i have an hypercube described by:
$$ C=\left\{ \vec{i}=(i_1,...,i_k) : ||\vec{i}||_{\infty} \leq n-1, i_k \in \left\{ 0,1,\ldots,n-1\right\},k=1,\ldots,n-1 \right\} $$
Let's consider
$$ C_1 = C \cap \left\{0\leq i_1 \leq i_2 \leq \ldots \leq i_k \right\}, $$
The question is: 1. Is it correct to say that all the edges, faces, other subset of greater dimension, are given by setting two or more indices to be equal? 2. For given object described in the previous point how many other hyper-tetrahedra share such an object?
What I've tried so far is to work on permutations/repetitons etc, but my argument is more based on observations rather than formal theory, and I'm pretty sure such problems are well known, i have some background in combinatorics algorithms if that could help a bit to formalize what i'm trying to do.
(I'm sorry for the formality, but actually combinatorics it's not my strong point).
