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 $$\sum_{\lambda : \lambda_1 \leq m} q^{a|\lambda|}P_{2\lambda}(1,q,q^2,\ldots ; q^{n}) = \text{"Infinite product modular function"}$$ 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 $q^{|\lambda|/2}P_{\lambda}(1,q,q^2,\ldots;q^n)$. 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.