Hall–Littlewood polynomials $P_\lambda(x;t)$ is an important deformation of Schur polynomials forming a basis in the ring of symmetric polynomials over $\mathbb Z[t]$. There are various definitions, including quite explicit $$ P_\lambda(x;t)= \sum_{w\in S_n/\lambda} w\left(x^\lambda\prod_{\lambda_i>\lambda_j}\frac{x_i-tx_j}{x_i-x_j}\right) $$ but I’m interested in a combinatorial description.
Schur polynomial $s_\lambda$ is a sum of monomials corresponding to semi-standard Young tableux of shape $\lambda$ — so I'm expecting an answer in the form of a weight on SSYT.
P.S. I'm mostly interested in the principal specialization of H-L polynomials — maybe this $(q,t)$-weight on SSYT is easier to describe than the full answer.
I'm not sure if this is exactly what you're looking for, but I recently worked on a paper regarding just this question (http://arxiv.org/abs/1403.8139). We were generalizing Tokuyama's deformation formula (http://projecteuclid.org/euclid.jmsj/1230129805) for the Schur polynomial, so we used Gelfand-Tsetlin patterns rather than Young tableaux. Tokuyama's original formula is in terms of strict GT patterns, which are in bijection with standard Young tableaux, but extending to the Hall-Littlewood polynomials required using nonstrict GT patterns (which are in bijection with SSYT).