I would like a good estimate for the number $ f(n)$ of functions $ \varphi\colon [n]\to 2^{[n]}$ , where $ [n]=\{1,2,\dots,n\}$ and $ 2^{[n]}$ is the set of all subsets of $ [n]$ , satisfying for all $ i\in [n]$ the two conditions $ $ \varphi(i) \subseteq \{i+1,i+2,\dots,n\} $ $ $ $ j\in \varphi(i) \Rightarrow \varphi(j)\subset \varphi(i). $ $ Simple bounds are $ $ 2^{\lfloor n/2\rfloor\lceil n/2\rceil} \leq f(n)\leq 2^{{n\choose 2}}. $ $ The problem arose from trying to find a simpler proof of the result of Kleitman and Rothschild that the number $ g(n)$ of nonisomorphic $ n$ -element partially ordered sets satisfies $ g(n)=2^{\frac 14n^2 +o(n^2)}$ .