Let $\Lambda$ be a Nakayama algebra and $\Lambda=\coprod\limits_{i=1}^sn_iP_i$ with $P_i$ nonisomorphic indecomposable projective modules ($n_iP_i$ denotes $n_i$ copies of $P_i$). Prove that the total number of nonisomorphic uniserial $\Lambda$-modules is $\sum\limits_{i=1}^s=l(P_i)$ where $l(P_i)$ is the length of $P_i$.
Evidentially, the deal is to show that there are no more uniserial modules except the ones contained in the unique composition series $0\subset Q_{i,1}\subset Q_{i,2}\subset\dots\subset P_i$ ($Q_{i,j}$ and $Q_{i,j+1}$ are nonisomorphic for each $j$) and that for $i\ne j$ the composition series for $P_i$ and for $P_j$ do not contain the same terms but I don't really know how to show this...
Can you please help me?