# Problem 553: Power sets of power sets
Let P(n) be the set of the first n positive integers {1, 2, ..., n}. Let
Q(n) be the set of all the non-empty subsets of P(n). Let R(n) be the
set of all the non-empty subsets of Q(n). An element X ∈ R(n) is a
non-empty subset of Q(n), so it is itself a set. From X we can construct
a graph as follows: Each element Y ∈ X corresponds to a vertex and
labeled with Y; Two vertices Y1 and Y2 are connected if Y1 ∩ Y2 ≠ ∅. For
example, X = {{1}, {1,2,3}, {3}, {5,6}, {6,7}} results in the following
graph: This graph has two connected components. Let C(n,k) be the number
of elements of R(n) that have exactly k connected components in their
graph. You are given C(2,1) = 6, C(3,1) = 111, C(4,2) = 486,
C(100,10) mod 1 000 000 007 = 728209718. Find
C(104,10) mod 1 000 000 007.