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.

  • 1
    $\begingroup$ Why was this interesting question downvoted? $\endgroup$ Jan 5, 2017 at 1:06

2 Answers 2


In a paper entitled $(q,t)$-analogues and $GL_n(\mathbb{F}_q)$, V. Reiner and D. Stanton have some such formulas.

Caveat. The above-mentioned $(q,t)$-binomial coefficients are a tad bit different from what you would call a "natural" binomial analogue. They comment that it is not even a polynomial (almost). Yet, the authors achieved very interesting outcomes with right-hand side mimicking what you quoted in your question. I let you decide the rest for yourself.


Siddhartha Sahi has some stuff on binomial coefficients and interpolation Macdonald polynomials, see e.g., this paper, where the qt-binomial is a quotient of certain specializations of interpolating Macdonald polynomials.

I am not sure if these satisfy an analogue of (1) thought.


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