Combinatorial approach to Andrews-Gordon and Bressoud type identities

We provide combinatorial tools inspired by work of Warnaar to give combinatorial interpretations of the sum sides of the Andrews-Gordon and Bressoud identities. More precisely, we give an explicit weight- and length-preserving bijection between sets related to integer partitions, which provides these interpretations. In passing, we discover the $q$-series version of an identity of Kur\c{s}ung\"oz, similar to the Bressoud identity but with opposite parity conditions, which we prove combinatorially using the classical Bressoud identity and our bijection. We also use this bijection to prove combinatorially many identities, some known and other new, of the Andrews-Gordon and Bressoud type.