Minimum chain partition of a Boolean lattice
= Minimum chain partition of a Boolean lattice
The least number of chains partitioning $\mathcal P([n])$ is the size of its largest rank:
$$
\binom n{\lfloor n/2\rfloor}.
$$
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.