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{\bf Martin Klazar}
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{\bf Counting Set Systems by Weight}
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Applying enumeration of sparse set partitions, we show that the number
of set systems $H\subset\exp(\{1,2,\dots,n\})$ such that
$\emptyset\not\in H$, $\sum_{E\in H} |E|=n$ and $\bigcup_{E\in H}
E=\{1,2,\dots,m\}$, $m\le n$, equals $(1/\log(2)+o(1))^nb_n$ where
$b_n$ is the $n$-th Bell number. The same asymptotics holds if $H$ may
be a multiset.  If the vertex degrees in $H$ are restricted to be at
most $k$, the asymptotics is $(1/\alpha_k+o(1))^nb_n$ where $\alpha_k$
is the unique root of $\sum_{i=1}^k x^i/i!-1$ in $(0,1]$.

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