Take the set $A_n=\{a_1,...,a_n\}$. Let $S_n$ be the set of subset-sums of $A_n$. (The subset-sum of the empty set is assumed to be zero.) Assume that there are $2^n$ unique members of $S_n$. How many possible sortings are there of set $S_n$?
For instance, if $n=2$, we have $S_2=\{0,a_1,a_2,a_1+a_2\}$. The number of possible sortings of $S_2$ is 8: $\{0<a_1<a_2<a_1+a_2, 0<a_2<a_1<a_1+a_2, a_2<0<a_1+a_2<a_1, a_1<0<a_1+a_2<a_2, a_2<a_1+a_2<0<a_1, a_1<a_1+a_2<0<a_2, a_1+a_2<a_1<a_2<0, a_1+a_2<a_2<a_1<0\}.$ $$\left\{ \begin{matrix} 0<a_1<a_2<a_1+a_2,\\ 0<a_2<a_1<a_1+a_2,\\ a_2<0<a_1+a_2<a_1, \\ a_1<0<a_1+a_2<a_2, \\ a_2<a_1+a_2<0<a_1, \\ a_1<a_1+a_2<0<a_2, \\ a_1+a_2<a_1<a_2<0, \\ a_1+a_2<a_2<a_1<0 \end{matrix} \right\}.$$
