**Here's the new, more thought through version.**

Consider a sequence of nonnegative integers $\lambda=(\lambda_1,\ldots,\lambda_n)$ with $\lambda_i\ge \lambda_{i+1}+2$ (the weight $\lambda-2\rho$ is dominant, in terms of $\mathfrak{gl}_n$ weights). Next, consider the polynomial $$P_\lambda(x_1,\ldots,x_n;t)=\sum\limits_{\sigma\in S_n} \sigma\left(x_1^{\lambda_1}\ldots x_n^{\lambda_n}\prod\limits_{i<j}\dfrac{x_i-tx_j}{x_i-x_j}\right).$$ This can, probably, be rightfully referred to as the Hall-Littlewood polynomial (all parts are distinct, so no normalization is needed). I'm just trying to accentuate the fact that I'm concerned with this specific polynomial, not an element of $\Lambda_{\mathbb{C}[t]}$. (Or is then "Hall-Littlewood polynomial" not the appropriate term?)

Anyway, I strongly believe that with our assumption on $\lambda$ in place the polynomial $P_\lambda(x_1,\ldots,x_n;-t)$ has positive coefficients. As I mentioned in the first version, this is confirmed by my observations and a certain geometrical argument.

My assumption can be somewhat weakened, but even as is this looks to me like a very basic fact in the theory of these well-studied expressions. Now, my questions are:

1) Can someone confirm that this is true and provide a reference to some down-to-earth (combinatorial) proof?

2) Why is this absent from all (almost all?) surveys on the subject of Hall-Littlewood polynomials? Just because this is a statement about the polynomials themselves rather than symmetric functions? Is it really not mentioned in Macdonald's book?

3) My real question. Is there a proof expressing $P_\lambda(x_1,\ldots,x_n;-t)$ as a sum of visibly positive summands enumerated by some combinatorial set (hopefully, SSYTs or Gelfand-Tsetlin patterns)?

andreduce to Hall-Littlewood poynomials at $q=0$? (I don't really know what I'm talking about, just trying to use my common sense.) $\endgroup$7more comments