Let $P(n)$ be the set of the first $n$ positive integers $\{1,2,\dots ,n\}$.
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 $X\in R(n)$ 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 $Y\in X$ corresponds to a vertex and labeled with $Y$;
• Two vertices $Y_1$ and $Y_2$ are connected if $Y_1\cap Y_2\ne \mathrm{\varnothing }$.

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 $C(2,1)=6$, $C(3,1)=111$, $C(4,2)=486$, $C(100,10)mod1000000007=728209718$.

Find $C({10}^4,10)mod1000000007$.