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

## TL;DR

This paper develops a graph-theoretic approach to derive Andrews--Gordon type generating function identities using $q$-difference systems and $q$-multi-summations, providing non-computer-assisted proofs linked to binary trees.

## Contribution

It introduces a novel method connecting graph theory, $q$-difference systems, and $q$-multi-summations to prove Andrews--Gordon identities without computational aid.

## Key findings

- Derived new Andrews--Gordon type identities
- Connected $q$-multi-summations with binary trees
- Provided non-computer-assisted proofs

## Abstract

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.