Sperner's theorem and the LYM inequality

Idea. A maximal chain adds one element at a time, so it corresponds to a permutation; an antichain meets each chain at most once.

LYM. There are $n!$ maximal chains. Each set $A$ of size $a$ lies on exactly $a!\,(n-a)!$ of them. Since $\mathcal F$ is an antichain, each chain contains at most one member of $\mathcal F$. Therefore
$$\sum_{A\in\mathcal F}|A|!\,(n-|A|)!\le n!.$$
Dividing by $n!$ gives $\sum_{A\in\mathcal F}\dfrac1{\binom n{|A|}}\le1$.

Sperner. For every $a$, $\binom na\le\binom n{\lfloor n/2\rfloor}$, so
$$\frac{|\mathcal F|}{\binom n{\lfloor n/2\rfloor}}\le\sum_{A\in\mathcal F}\frac1{\binom n{|A|}}\le1.$$
∎

Sharpness. All subsets of size $\lfloor n/2\rfloor$ form an antichain of that size.

Lesson. Double count (set, maximal chain) pairs: chains correspond to permutations and give clean factorial weights.