Transition matrices for symmetric and quasisymmetric Hall-Littlewood polynomials

We introduce explicit combinatorial interpretations for the coefficients in some of the transition matrices relating to skew Hall-Littlewood polynomials $P_{\lambda / \mu}(x;t)$ and Hivert's quasisymmetric Hall-Littlewood polynomials $G_{\gamma}(x;t)$. More specifically, we provide the following: 1. $G_{\gamma}$-expansions of the $P_{\lambda}$, the monomial quasisymmetric functions, and Gessel's fundamental quasisymmetric functions $F_{\alpha}$, and 2. an expansion of the $P_{\lambda / \mu}$ in terms of the $F_{\alpha}$. The $F_{\alpha}$ expansion of the $P_{\lambda / \mu}$ is facilitated by introducing the set of $\textit{starred tableaux}$. In the full version of the article we also provide $G_{\gamma}$-expansions of the quasisymmetric Schur functions and the peak quasisymmetric functions of Stembridge.