I'm interested in the set $F_n$ of functions $f:\{-1,1\}^n\to\{-1,1\}$ which can be represented as $x\mapsto\text{sign}(a\cdot x)$ for some $a\in\mathbf{R}^n$. What is the size of $F_n$? Is there some discrete encoding which is 'better' then to write down the complete lookup table?
My idea is to look at the number of connected components in $\mathbf{R}^n\setminus(\cup_x\{x\}^\bot)$, since if $a,a'$ represent different functions $f,f'$, say $f(y)\neq f'(y)$, then I can't go from $a$ to $a'$ without crossing the border $a\cdot y=0$. But I still don't know how many such components there are.
[EDIT] I just tried to plot the planes $\{x\}^\bot$ in $\mathbf{R}^3$ (w.l.o.g. $x_1=1$ since $\{x\}^\bot=\{-x\}^\bot$), there seem to be $14$ partitions and they seem to correspond to the faces and edges of the $[-1,1]^3$-cube. But in even dimensions the picture needs to be different since the edges of the $[-1,1]^{2n}$-cube lie directly on the planes
EDIT: My previous rough idea about orthants was completely off of the mark. I need to go rethink everything.
I have provided a link to a set of slides in the comments below, but I think I'm starting to get out of my depth. Someone who knows better than I should swoop in and save the day.