Lubell--Yamamoto--Meshalkin inequality

If $\mathcal{A}$ is an antichain in $\mathcal{P}([n])$ and ${\mathcal{A}}_h$ is its rank-$h$ part, thenThe left side is the expected number of members of $\mathcal{A}$ on a Uniformly random maximal chain in a Boolean lattice.

For $\mathcal{A}\subseteq [n{]}^{(r)}$, the local LYM inequality saysIt follows by counting pairs $(B,A)$ with $B\subset A$, $|B|=r-1$, and $A\in \mathcal{A}$.

Every antichain in $\mathcal{P}([n])$ has at most $\left(\genfrac{}{}{0ex}{}{n}{\lfloor n/2\rfloor }\right)$ members. Equality is attained only by a full middle level, with either middle level possible when $n$ is odd.