  Finding an Andrews--Gordon type generating function identity for a linked
partition ideal is difficult in most cases. In this paper, we will handle this
problem in the setting of graph theory. With the generating function of
directed graphs with an ``empty'' vertex, we then turn our attention to a
$q$-difference system. This $q$-difference system eventually yields a
factorization problem of a special type of column functional vectors involving
$q$-multi-summations. Finally, using a recurrence relation satisfied by certain
$q$-multi-summations, we are able to provide non-computer-assisted proofs of
some Andrews--Gordon type generating function identities. These proofs also
have an interesting connection with binary trees.
