  We introduce the notion of a directed stratification for a finite-dimensional
algebra. For algebras that admit such a stratification we characterise the
projective resolutions of finitely generated modules and obtain a result for
the finitistic dimension, which is an inductive version of a result of Fossum,
Griffith and Reiten. With the developed techniques, which are adopted from the
theory of EI-category algebras, we gain deeper insight in the combinatorial
nature of this result. A characterisation of algebras which do not admit a
directed stratification is given in terms of the Ext-quiver.
