Equality in Sperner theorem ($|\mathcal{A}|=\left(\genfrac{}{}{0ex}{}{n}{\lfloor n/2\rfloor }\right)$)

The maximum antichains in the Boolean lattice are exactly its full middle levels. For even $n$ there is one such level, and for odd $n$ either middle level works. The LYM inequality first confines a maximum antichain to those levels; connectedness of the regular inclusion bipartite graph rules out a mixture of the two odd middle levels.