$(q,x)$-analog of $n!$ - MathOverflow

$(q,x)$-analog of $n!$

2012-02-01T04:02:42Z

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.

Answer by Gjergji Zaimi for $(q,x)$-analog of $n!$

2012-02-01T07:48:23Z

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 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.