## Power sets of power sets

### Problem 553

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
`Y`_{1}and`Y`_{2}are connected if`Y`_{1}∩`Y`_{2}≠ ∅.

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`(10^{4},10) mod 1 000 000 007.