  In his paper \cite{MR1}, Markus Reineke proposed a conjecture that there
exists a stable weight system $\Theta$ for every indecomposable representation
of Dynkin type quiver. In this paper, we showed this conjecture is true for
quivers of $A_n$-type by combinatorial construction of a special weight system.
We also reinterpret this weight system in terms of semi-invariant theory.
