Consider the standard embedded $n$-cross polytope $P_n$ with vertices $\pm e_i \in \mathbb R^n$. Let us consider decompositions of this polytope into $2^{n-1}$ simplices, such that these simplices have only vertices that are also vertices of $P_n$. How many different such decompositions exist? (Note that such a decomposition is not necessarily a triangulation.)

Secondly, I want to know the same about the $n$-polytope $Q_n$ which is defined to be the convex hull of the set $\{e_i+e_j \in \mathbb R^{n+1} \ | \ 1 \leq i < j \leq n+1\}$. (Think about $Q_n$ as a $n$-simplex which is twice as large as usual and has its corners removed.)