Let $Q_n$ denote the $n$-dimensional hypercube graph. I am interested in counting the number of ways to partition $Q_n$ into $2^{n-d}$ vertex-disjoint copies of $Q_d$, for some integer parameter $1 \leq d \leq n$.
For example, when $n = 2$ and $d=1$, then the answer should be 2. We label the vertices of $Q_2$ by $\{00, 01, 10, 11\}$. Then we can partition $Q_2$ as $\{00, 01\}$ and $\{10, 11\}$, which are two vertex disjoint copies of $Q_1$. Alternatively, the partition $\{00, 10\}$ and $\{01, 11\}$ is also valid for the same reason -- hence giving the answer of 2.
Is there a way to establish a recurrence for this problem in terms of $n$ and $d$ perhaps?