TITLE: Above/below directed graph on cells of arrangement of lines
QUESTION [7 upvotes]: This question concerns the structure of a directed graph
built on the cells of an arrangement of lines.
My basic question is whether this graph has been
studied before, perhaps in another guise. I also ask a specific
question.

Let $\cal A$ be an arrangement of $n$ lines in $\mathbb{R}^2$,
with no line vertical, and no two lines parallel.
I associate to $\cal A$ a directed graph $G$,
with each node a cell of $\cal A$,
and $a \to b$ iff cell $a$ is adjacently above $b$
in the sense that (1) $a$ and $b$ share an edge $e$,
and (2) for a point $p$ in the interior of $e$,
points vertically just below $p$ lie in $b$, and points
vertically just above $p$ lie in $a$.
Thus $e$ is an upper edge of $b$ and a lower edge of $a$.
Let $L^+$ be the line of $\cal A$ with the largest positive slope,
and $L^-$ the line with the steepest negative slope.
(From the assumptions, $L^+$ and $L^-$ are unique and distinct.)
Then there is a unique unbounded cell $A$ of $\cal A$ that is
bounded by $L^+$ and $L^-$ and has no cell adjacently above it,
and similarly there is a unique unbounded cell $B$, also
bounded by $L^+$ and $L^-$, which has no cell adjacently below.
So $A$ is the unique source of $G$, and $B$ the unique sink.
Now I assign numbers/weights to the nodes of $G$ as follows.
$A$ is assigned $1$.
For any node $x$ of $G$, $x$ is assigned the sum of the
weights of the nodes adjacently above $x$,
i.e., with incoming arcs to $x$.
Every node but $A$ has at least one incoming arc, and so
this is well-defined.
Two $4$-line arrangements are shown below.

         


         

$L^+$ and $L^-$ are colored green.


Every node but $B$ has at least one outgoing arc, so the
weight of a node is always propagated downward.
Thus the sink node $B$ receives the highest weight.
In (a), $B$ has weight $8$, whereas in $B$
it has weight $9$.
A basic question is:

Q1. What is the largest possible weight assigned to the sink node $B$,
  over all arrangements of $n$ lines?

Here is a more complicated arrangement of $8$ lines, whose
sink node has weight $111$.

         


(This was calculated by hand, so apologies if there are errors.
Thanks to Will Brian for catching one error.)
The more general question is:

Q2. Has this graph and node-weighting scheme been studied
  before, perhaps in another context?

REPLY [2 votes]: We can always achieve exactly $2^{n-1}$ for the weight of the sink. This gives us a lower bound for the answer to Q1.
Essentially, the idea is that we draw our $n$ lines all tangent to the upper half of a circle. We then obtain a weight pattern identical to the first $n$ rows of Pascal's triangle, and the weight of the sink is gotten by adding together all the entries in the bottom row. I'm giving only a crude explanation, because a picture explains it much better. Here is $n = 5$ (the circle is shown in gray, and the faded red lines are showing you the rows of Pascal's triangle):