Symmetric chain decomposition of a Boolean lattice

A symmetric chain in the Boolean lattice $\mathcal{P}([n])$ contains one set of each size from $r$ through $n-r$. A symmetric chain decomposition partitions the Boolean lattice into such chains.

A symmetric chain contains one subset of each size from $k$ to $n-k$, nested by inclusion. Its endpoint ranks sum to $n$. A symmetric chain decomposition of a Boolean lattice partitions all subsets into such chains.

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.

The least number of chains partitioning $\mathcal{P}([n])$ is the size of its largest rank:The middle rank gives the lower bound because a chain contains at most one member of any rank. Compatible adjacent-level matchings give a symmetric chain decomposition attaining the bound.