  Motivated by the structure of the algebras associated to the blocks of the
BGG-category O we define a subclass of quasi-hereditary algebras called
1-quasi-hereditary. Many properties of these algebras only depend on the
defining partial order. In particular, we can determine the quiver and the form
of the relations. Moreover, if the Ringel dual of a 1-quasi-hereditary algebra
is also 1-quasi-hereditary, then the structure of the characteristic tilting
module can be computed.
