# Decomposition of a cross-polytope into simplices

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.)

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. 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).

The $n$-dimensional cross-polytope $P_n$ can be triangulated (divided into simplices, without adding vertices) in exactly $n$ different ways. Each of these has $2^{n-1}$ simplices.
Much less seems to be known about the triangulations of $Q_n$, the "rectified $n$-simplex".