Let $\mathbf{F}$ be a family of subsets of N = {1,2,...,n}, and suppose there are no $A, B \in \mathbf{F}$ satisfying $A \subset B$. Let $\delta \in S_n$ be a random permutation of the elements of N and consider the random variable $$[ X = |\{i:\{\delta(1),\delta(2)),\dots,\delta(i)\}\in \mathbf{F}\}| ]$$
By considering the expectation of X prove that $|\mathbf{F}| \leq (^{ n}_{\lfloor n/2 \rfloor})$.
Found a proof that And we can easily know that $E[X]$ is either 0 or 1, as for each permutation, if $\delta=\{\delta(1),\delta(2)),\dots,\delta(i)\}$ in F, then any t that $t \neq i$, $\{\delta(1),\delta(2)),\dots,\delta(t)\}$ should not in $\mathbf{F}$. So $E[X] \leq 1$. Let $A_i$ be the number of elements of F with size i, $E[X]=\sum_i \frac{A_i}{(^n_i)} \geq\sum_i \frac{A_i}{(^{\quad n}_{\lfloor n/2 \rfloor})} $, so $|\mathbf{F}| \leq (^{\quad n}_{\lfloor n/2 \rfloor})$.
But I do not know why $E[X]=\sum_i \frac{A_i}{(^n_i)}$.