Consider a connected skew Young diagram in the English notation and then rotate it counterclockwise by $\pi/4$. This rotation can be avoided by simply replacing "rows" by "diagonals" in the below, but I'm used to looking at it this way. (This is because the fillings I'm interested in satisfy Gelfand-Tsetlin type inequalities for which such an orientation is standard. That being said, fillings do not appear explicitly in my question.)

Anyhow, let a row be the set of our diagram's squares (the centers of) which lie on the same horizontal line. We're then interested in the polynomial $$\prod (1-t^i)^{d_i},$$ where $d_i$ is the number of pairs of consecutive rows, such that the upper one contains $i-1$ squares and the lower one contains $i$ squares. (The top row contains one square and provides a factor of $(1-t)$.)

These expressions appear in the standard combinatorial formula for Hall-Littlewood polynomials, the one that follows directly from the branching rule (found in Macdonald).

My question is: can anyone provide some insight on what these polynomials really are? In other words I'm looking for some different more natural definition. Is it some kind of characteristic polynomial? Some determinant? Some probability when $0<t<1$? Anything at all?

This is, probably, pretty obscure unless you know the answer... I ask since these expressions figure prominently in my work and have interesting properties for which I know no transparent explanation. It would also be very interesting to know if someone has come across them in some other context.