The $q$-binomial coefficients ${n + k \choose n}_q$ can be written as a sum

$${n + k \choose n}_q = \sum_{\lambda \subset (n^k)} q^{|\lambda|}\qquad (1)$$

where $(n^k) = (n, n, ..., n)$ is a partition of $k$ parts each part equal to $n$. For a proof see e.g. Theorem 2 of this book chapter.

Question: Is there a $(q, t)$-generalisation of (1) that is related to the Macdonald polynomials in whatever sense?

Below is what I've found after searching on the Internet:

arXiv:1001.3466 and arXiv:q-alg/9605004 both introduced certain $(q, t)$-analogues of the binomial coefficients. Do they satisfy any analogues of (1)?

Also, in this book chapter the $(q, t)$-Catalan numbers have a nice lattice path interpretation analogous to (1), see Formula (83) in that chapter. But there does not seem to be any $(q, t)$-binomial coefficients there.