Parity considerations in Andrews–Gordon identities

In 1974, Andrews discovered the generating function for the partitions of n considered in a theorem due to Gordon. In a more recent paper, he reconsidered this generating function and gave refinements where additional restrictions involving parities are imposed. A combinatorial construction for the partitions enumerated by the mentioned generating function is given. Some of the Andrews’ refinements are proven combinatorially, and a conjecture of his is settled.