A chain $C_1 \subseteq C_2 \subseteq ... \subseteq C_m$ in P(n) is symmetric if $|C_{i+1}| = |C_i| +1$, for all $i=1, ..., .m-1$, and $|C_1| + |C_m| = n$.

Sperner's lemma shows that there is a partition of the power set into $\binom{n}{\lfloor n/2 \rfloor}$ chains. But these could be asymmetric.

Proposition: For $n\geq 1$, there is a partition of P(n) into symmetric chains.