Inequality on the family of intersecting antichain Let $\{A_1, A_2,..., A_m\}$ be an intersecting antichain of subsets of $[n]$ s.t. $|A_i| \leq n/2$ for each $i$. Prove that
$$\sum_{i=1}^m \binom{n-1}{|A_i|-1}^{-1} \leq 1.$$
I know it originally appeared in Sperner Systems Consisting of Pairs of Complementary Subsets [Bollobas, 1972], but the proof in the paper is somehow confusing to me. Could anyone give me more explanations on that? Thanks a lot!

I think I can answer it myself now.
For $A \in \mathcal{F}$ and $\pi$, a permutation of $[n]$. We let $w(\pi, A) = 1/|A|$ if $\pi(A)$ is cyclically consecutive and let $w(\pi, A) = 0$ otherwise. We do double counting on $\sum_{\pi, A} w(\pi, A)$. First we fix $A$ and we have
$$\sum_{\pi, A} w(\pi, A) = \sum_{A \in \mathcal{F}} = \sum_A n|A|!(n-|A|)!/|A|.$$
Then we fix $\pi$ and by the Lemma in the paper we have
$$\sum_{\pi, A} w(\pi, A) = \sum_\pi \sum_{A \in \mathcal{F}} w(\pi, A) \leq \sum_\pi 1 = n!,$$
completing the proof.
