Generalized Andrews-Gordon Identities

In a recent paper, Griffin, Ono and Warnaar present a framework for Rogers-Ramanujan type identities using Hall-Littlewood polynomials to arrive at expressions of the form $$\sum _{\lambda :{\lambda }_1\le m}{q}^{a\mathrm{\mid }\lambda \mathrm{\mid }}{P}_{2\lambda }(1,q,q^2,\dots ;q^n)=\text{"Infinite product modular function"}$$ for $a=1,2$ and any positive integers $m$ and $n$. A recent paper of Rains and Warnaar presents further Rogers-Ramanujan type identities involving sums of terms $q^{\mathrm{\mid }\lambda \mathrm{\mid }\mathrm{/}2}{P}_{\lambda }(1,q,q^2,\dots ;q^n)$. It is natural to attempt to reformulate these various identities to match the well-known Andrews-Gordon identities they generalize. Here, we find combinatorial formulas to replace the Hall-Littlewood polynomials and arrive at such expressions.