  To determine an algebra is quasi-hereditary is a difficult problem. An
effective method, Green-Schroll set, is introduced in this paper to tackle this
problem. It is well known that an algebra is quasi-hereditary if and only if it
admits a quasi-hereditary ordering of simple modules. Let $A$ be a Nakayama
algebra. We prove a necessary and sufficient criterion to determine whether an
ordering of simple modules is quasi-hereditary on $A$, and $A$ is
quasi-hereditary if and only if its Green-Schroll set is nonempty. This seems
to be the simplest characterization currently known, since it does not involve
any algebraic concepts. A general iteration formula for the number $q(A)$ of
all quasi-hereditary orderings of $A$ is given via Green-Schroll set. The
$q$-ordering conjecture is proved to be true for $A$.
