co.combinatorics – Refined Inverse Plane Partition Generation Function


I have a simple question about the generation function for inverse plane partitions:

$$ sum _ { pi in RPP ( lambda)} z ^ {| pi |} = prod_ {s in lambda} frac {1} {1-z ^ {h _ { lambda} (s)}} $$

There is a natural refinement of the right side:

$$
prod_ {s in lambda} frac {1} {1-t z_1 ^ {a _ { lambda} (s)} z_2 ^ {l _ { lambda} (s)}}
$$

Or maybe just with $ t = z_1, z_2 $.

I suspect that there should be a right side equivalent to this identity, that is, to count some "refined weight" of the inverse plane partition. Maybe along diagonals? I was wondering if there is any known generating function. I think there is a natural geometric interpretation as a Poincar polynomial in $ z_2 $.

If it helps, the right side is something like $ c _ lambda (q, t) $ of Macdonald's polynomial theory.

Thank you