Counting Set Systems by Weight

Applying enumeration of sparse set partitions, we show that the number of set systems H ⊂ exp({1, 2, . . . , n}) such that ∅ 6 ∈ H, ∑E∈H |E| = n and ⋃E∈H E = {1, 2, . . . ,m}, m ≤ n, equals (1/ log(2)+ o(1))bn where bn 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/αk + o(1))bn where αk is the unique root of ∑k i=1 x i/i! − 1 in (0, 1].