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Here we consider the $q$-series coming from the Hall-Littlewood polynomials,\begin{equation*} R_\nu(a,b;q)=\sum_{\substack{\lambda \\[1pt] \lambda_1\leqa}} q^{c|\lambda|} P_{2\lambda}\big(1,q,q^2,\dots;q^{2b+d}\big).\end{equation*} These series were defined by Griffin, Ono, and Warnaar in theirwork on the framework of the Rogers-Ramanujan identities. We devise a recursivemethod for computing the coefficients of these series when they arise withinthe Rogers-Ramanujan framework. Furthermore, we study the congruence propertiesof certain quotients and products of these series, generalizing the famousRamanujan congruence \begin{equation*} p(5n+4)\equiv0\pmod{5}. \end{equation*}

Combinatorial Properties of Rogers-Ramanujan-Type Identities Arising from Hall-Littlewood Polynomials