I have a simple question about the generating function for reverse plane partitions:

$$\sum_{\pi \in RPP(\lambda)} z^{|\pi|}= \prod_{s \in \lambda} \frac{1}{1-z^{h_{\lambda}(s)}}$$

There's a natural refinement of the right hand side:

$$ \prod_{s \in \lambda} \frac{1}{1-t z_1^{a_{\lambda}(s)}z_2^{l_{\lambda}(s)}} $$

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

I suspect there should be an equivalent left hand side to this identity - i.e. counting some "refined weight" of the reverse plane partition. Perhaps along diagonals? In a sense there has to be - I'm just not sure what the "statistic" is to count. I wondered if there is a known generating function?

If it helps the right hand side is something like $c_{\lambda}(q,t)$ from Macdonald polynomial theory.

Update: If I write this instead in terms of $w_1 = z_1/z_2$ and $w_2 = z_1z_2$ I actually expect the RHS to be a polynomial in $w_1$ as in something along the lines:

$$ \sum_{\pi \in RPP(\lambda)} w_2^{|\pi|}P_{|\pi|}(w_1) $$

Where $P(w_2)$ is a finite polynomial.

Thanks

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