As David Speyer points out in a comment, this is equivalent to the number of regions resulting when $\mathbb{R}^n$ is divided by hyperplanes of the form $\sum_{i \in I}a_i = \sum_{i \in J}a_i$ for all disjoint pairs of subsets $I,J \subseteq [n]$. Dual to this description, it is the number of ways that the $3^n-1$ nonzero vectors in $\{-1,0,1\}^n$ can be divided into "positive" and "negative" by a hyperplane (in general position) passing through the origin, which makes it easy to see that the answer is indeed $8$ for $n=2$.

As I mentioned in a comment, the total is always $2^n$ times what you get when making the assumption $a_i > 0$ for all $i$. There is another symmetry that can also be exploited: making the assumption $a_1 < a_2 < \cdots < a_n$ reduces the total by a further factor of $n!$, which gives a known sequence:

http://www.oeis.org/A009997

http://arxiv.org/abs/math.CO/9809134

Various key words are "coherent boolean term order", "coherent generalized term order", and "additive antisymmetric comparative probability order". It doesn't look like anyone knows the values beyond $n=7$. You'll want to check Maclagan's reference to Fine and Gill 1976 to see if they give any asymptotics.

Including the $2^nn!$ symmetries gives these values:

- $2$
- $8$
- $96$
- $5$ $376$
- $1$ $981$ $440$
- $5$ $722$ $536$ $960$
- $138$ $430$ $238$ $607$ $360$

`$\mathbb{R}^n$`

is divided into by the hyperplanes of the form`$\sum_{i \in I} a_i = \sum_{j \not \in I} a_j$`

. I feel like this example should be in some survey on combinatorics of hyperplane arrangements, but I don't know which one. $\endgroup$ – David E Speyer Sep 20 '10 at 13:53