Let U be the set $U=\{(1,2,3,\ldots,2^m)\}$. Let $A$ and $B$ partitions of $U$, such that $A \cup B$ is the set $U$, and their intersection is empty, and adding the elements of the first set is the same number of the addition of the elements of the second set $B$.
How many $A$ and $B$ is there?
I mean, if $U=\{1,2,3,4\}$, there exist just a single pair of sets $A$, and $B$. $A=(3,2)$ and $B=(4,1)$, because $3+2=4+1$, i want to know how many $A$ and $B$s exists for bigger sets $U$.