Symmetric Chain Decomposition

[YaelDillies]

For every $n$, $2^{[n]}$ can be decomposed into symmetric chains where a chain is symmetric if it is of the form $C_i \subseteq \dots \subseteq C_{n - i}$ with $|C_j| = j$ for all $i \le j \le n - i$.

This is Theorem 1.11 in the lecture notes. There's a start at #21.