Number of ordered set partitions I have encountered a counting problem which involves counting all ordered partitions.
I give an example: Given a set of two elements $\{A,B\}$, there are two partitions, $\{\{A,B\}\}$ and $\{\{A\},\{B\}\}$, but there are three ordered partitions, $(\{A,B\})$, $(\{A\},\{B\})$, and $(\{B\},\{A\})$.
I have to count them for a set of order $2^N$ with $N$ growing.
I am aware of the Stirling number $k\brace r$ of the second kind, which describes the number of $k$-partitions of a set with cardinality $r$. 
So I get some thing like this
$$ \sum_{k=0}^{2^n} k!{{2^n}\brace k} $$
Are there closed formulas for this problem? Is this problem wellstudied in combinatorics? Are asymptotic formulas known?
I am happy for any advice or reference I can get.

These are the ordered Bell numbers (or Fubini numbers)
$$a(n)=\sum_{k=0}^nk!{n\brace k}\;;$$
asymptotically they satisfy
$$a(n)\approx\frac{n!}{2(\ln 2)^{n+1}}\;.$$
The sequence of these numbers is OEIS A000670, where you’ll find many references and much information; there don’t seem to be any nice closed forms. You’re interested specifically in $a(n)$ when $n$ is a power of $2$; it does not appear to me that this helps.
