Fredrick puzzled:
> Is there a graph for every possible monotonic sequence?
No, there's not one for
\[ 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, \dots \]
I leave it as a puzzle to figure out why!
I don't think "monotonic" is especially relevant here. For example, it's pretty easy to find graphs where the number of paths from \\(x\\) to \\(x\\) goes like
\[ 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, \dots \]
or
\[ 1, 0, 0, 2, 0, 0, 4, 0, 0, 8, 0, 0, 16, 0, 0, \dots \]