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.)



Decomposition of a cross-polytope into simplices
The cross-polytope has $2n$ vertices in $n$ antipodal pairs. A simplex with $n+1$ points from these must contain exactly one antipodal pair of points, since if it contains more than one, it has a square face and $0$ volume. So, the simplex is the union of two adjacent cones over facets, or the intersection of the cross-polytope with two adjacent orthants. This means a division of the cross-polytope into $2^{n-1}$ simplices corresponds to a perfect matching of the dual hypercube. These are counted by [A005271](http://oeis.org/A005271). The first $7$ values are known. The number increases extremely rapidly, doubly exponentially, since one lower bound is $2^{2^{n-2}}$ (matchings contained in $2^{n-2}$ parallel squares) and one upper bound is $n^{2^{n-1}} = 2^{(\log\_2 n) 2^{n-1}}$ (choose a direction for each odd vertex of the cube). 


