I would like to know how one would triangulate an associahedron $K_n$ of any given dimension. The reason for this is that I was trying to show that the singular chains of an $A_\infty$-space (defined in Homotopy associativity of H-spaces) are an $A_\infty$-algebra. This a theorem in Homotopy associativity of H-spaces II by Stasheff (although he doesn't specify that he's using singular chains, he could be referring to cellular chains for instance), but the author doesn't provide any details, he just says that it's easy to verify up to sign, and then one has to take care of orientations.
I was able to show the lowest equations defining an $A_\infty$-algebra (the ones involving only $m_1,m_2,m_3$ and $m_4$). As Stasheff says, one has to choose a suitable generator of $C_*(K_n)$. For $K_2$ and $K_3$, since they are already $\Delta^0$ and $\Delta^1$ I would just choose the identity map from the correspoding simplex. For $K_4$, which is a solid pentagon, I would triangulate it so that the generator is a sum of 3 maps $\Delta^2\to K_4$, and then give suitable orientation so that the signs match.
However, I don't know how to triangulate higher dimensional associahedra, let alone giving then the correct orientations. So my main question is how to do this.
I also understand that since Associahedra are defined as cell complexes, it could be easier to use cellular chains, but I am not so well versed in this homology theory, so I would appreciate some guidance for showing the theorem in this setting if that makes it easier.