Linked partition ideals, directed graphs and q-multi-summations

In this paper, we start by considering generating function identities for linked partition ideals in the setting of basic graph theory. Then our attention is turned to q-difference systems, which eventually lead to a factorization problem of a special type of column functional vectors involving q-multi-summations. 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. Further, we give illustrations of constructing a linked partition ideal, or more loosely, a set of integer partitions whose generating function corresponds to a given set of special q-multi-summations.