Adjacent-level matching in a Boolean lattice

For $1\le i\le n/2$, the inclusion graph between ranks $i-1$ and $i$ of $\mathcal{P}([n])$ has a matching saturating rank $i-1$. Indeed, each lower vertex has degree $n-i+1$, each upper vertex has degree $i$, and edge counting givesfor every family $\mathcal{A}$ in the lower rank. This verifies the condition of the Hall marriage theorem.