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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.

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I'm not sure, is $W(m) = 0.5 N! m$ ? – user9292 Feb 15 '13 at 6:37
This sounds like homework. Please see our homework policy. We expect homework problems to have some effort put into them, and deal with conceptual issues. Could you please edit your question to explain (1) What you have tried, (2) the concept you have trouble with, and (3) your level of understanding? – Michael Brown Feb 15 '13 at 6:46
I'll give you a hint though. The maximum distance you could get for a given $N$ is $m=N$, so that function clearly doesn't work: it is unbounded in $m$. Try taking some small values of $N$ and $m$ where you can write out all the combinations. Write out all the sequences of length $N$ like LRRLRLLL... etc. and see what the pattern is for paths of length $m$. – Michael Brown Feb 15 '13 at 6:47
This is either a brilliant way to approach the stat mech of a spin-1/2 ensemble or a terribly confusing way - I can't decide which. – Chris White Feb 15 '13 at 6:57
How about $\frac{(m-N)!}{m!}$ – user9292 Feb 15 '13 at 7:10

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