Let us consider a square grid, which has been rotated by 45deg. On this grid we deﬁne a path, the directed polymer, which starts at the origin ($t = 0$) and extends in the positive $t$-direction (at each grid point the path goes either left or right; and steps in the negative $t$-direction are not permitted). All paths of length $N$ end at the same $t$-position. The distance of the endpoint to the $t$-axis is characterized by a number $m$. Each path represents a microstate and the endpoint (i.e. $m$) the macrostate.
How can I find the number of microstates, say, $W(m)$?
I've tried that $W(m) = 0.5 N! m$, but I'm not sure. I do understand the problem, but I'm not sure how to solve it.