Number of possible rod cuts of a long rod .

I take it you are only allowed to cut the rod into integer lengths. I don't know about dynamic programming, but if you mark all the places where you are allowed to cut the rod, there are $n-1$ of them, and at each of those $n-1$ places, either you cut, or you don't, making, all told, $2^{n-1}$ different ways to cut.