5658: Sereja and Sets

Let's assume that set S consists of m distinct intervals [l₁,r₁], [l₂,r₂], ..., [l_m,r_m] (1≤l_i≤r_i≤n; l_i,r_i are integers).
Let's assume that f(S) is the maximum number of intervals that you can choose from the set S, such that every two of them do not intersect. We assume that two intervals, [l₁,r₁] and [l₂,r₂], intersect if there is an integer x, which meets two inequalities: l₁≤x≤r₁ and l₂≤x≤r₂.
Sereja wonders, how many sets S are there, such that f(S)=k? Count this number modulo 1000000007 (10⁹+7).

3 1

23

3 2

32

2 0

1

2 2

2