While doing some work in geometric representation theory I have come across the following
sequence of polynomials in two variables $(q,x)$ which I would like to denote 
by $n!\_{q,x}$. For small $n$ these polynomials look as follows:


$2!\_{x,q}=x+q$


$3!\_{x,q}=x^3+x^2(q^2+q)+x(q^2+q)+q^3$


$$
4!\_{x,q}=x^6+x^5(q^3+q^2+q)+x^4(q^4+q^3+2q^2+q)+x^3(q^5+q^4+2q^3+q^2+q)+
$$
$$
x^2(q^5+2q^4+q^3+q^2)+x(q^5+q^4+q^3)+q^6
$$


The polynomials are actually symmetric in $q$ and $x$ and when one puts $x=1$ one recovers
the usual $q$-analog of $n!$ (in particular, when both $q$ and $x$ are 1, we get $n!$).


My question is this: has anybody seen such polynomials before? What is the correct definition of those polynomials for general $n$? Any information will be greatly appreciated.



$(q,x)$-analog of $n!$
I was hesitating to write an answer since I don't have references at hand but let me mention that if you denote your polynomials $P\_n(x,q)$ and look at $Q(x,q)=x^{\binom{n}{2}}P\_n(x^{-1},q)$ then (my guess is that) you are looking at:
$$Q(x,q)=\sum\_{\pi\in S\_n}x^{maj(\pi)}q^{inv(\pi)}=\sum\_{\pi\in S\_n}x^{maj(\pi)}q^{maj(\pi^{-1})}$$
where $maj(\pi)$ is the [Major index](http://en.wikipedia.org/wiki/Major_index) of a permutation
$$maj(\pi)=\sum\_{\pi(i) > \pi(i+1)}i.$$
These polynomials satisfy the following
$$
\sum \_{n=0} ^{\infty}\frac{Q \_n(x,q)}{(x) \_n(q) \_n}u^n = \prod \_{i,j=0} ^{\infty}\frac{1}{1-x^iq^ju} $$
where $(q)\_n$ denotes $(1-q)(1-q^2)\cdots(1-q^n)$. From this relation you can get a useful recurrence relation or other properties which may help you compute these polynomials.


