# What is the size of these discrete function sets?

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

• Hey, thanks a lot for the input! I just tried to plot the planes $\{x\}^\bot$ in $\mathbf{R}^3$ (w.l.o.g. $x_1=1$ since $\{x\}^\bot=\{-x\}^\bot$), but the resulting components don't look like orthants to me, some of them are enclosed by four planes but some only by three. Or am I confused about this? – fweth Dec 2 '18 at 3:25
• @fweth I found an interesting set of slides about counting regions of $\mathbb{R}^n$ that have been separated by hyperplanes. They even discuss what to do with hyperplanes that are not in general position, which is what we have here. It also looks a bit complicated. physics.csusb.edu/%7Eprenteln/papers/hyperintro.pdf – FranklinBash Dec 2 '18 at 4:21