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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.

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2  
These are the bivariate generating functions of number of (co)inversions and major index of permutations. –  Gjergji Zaimi Feb 1 '12 at 6:35
    
Thanks! Can I ask you a stupid question: why is it obvious that it is symmetric with respect to $q$ and $x$? –  Alexander Braverman Feb 1 '12 at 7:20
7  
It's not a stupid question. Note that these are also the bivariate generating functions recording $p^{maj(\pi^{-1})}q^{maj(\pi)}$, where the symmetry is more obvious (a combinatorial proof of this was given by Foata and Schutzenberger). I recommend reading "Permutation statistics and partitions" by Garsia and Gessel, or Gessel's thesis. Available here people.brandeis.edu/~gessel/homepage/papers/index.html –  Gjergji Zaimi Feb 1 '12 at 7:28
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To give credit to the original sources, these polynomials (and the combinatorial interpretation in terms of the major index) were first studied in B. Gordon, Two theorems on multipartite partitions. J. London Math. Soc. 38 (1963) 459-464. However Gordon did not explicitly describe the combinatorial interpretation that he found in terms of the major index. This was done in D. P. Roselle, Coefficients associated with the expansion of certain products. Proc. Amer. Math. Soc. 45 (1974), 144-150. –  Ira Gessel Feb 1 '12 at 15:09

1 Answer 1

up vote 15 down vote accepted

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.

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Thanks a lot - that's exactly what I need! –  Alexander Braverman Feb 1 '12 at 18:04
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For an extension to more than two variables and some references, see Exercise 4.47 of Enumerative Combinatorics, vol. 1, 2nd ed. –  Richard Stanley Feb 2 '12 at 2:10
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Continuing my previous comment, see also Corollary 7.23.9 of vol. 2 for a development in the context of symmetric functions. –  Richard Stanley Feb 2 '12 at 14:15

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