Let $

t_{a,b}$ be the numbers

$$

t_{a,b} := |{ pi in S_{a+b} : mathrm{maj}(pi)=a text{ and } mathrm{maj}(pi^{-1})=b }|.

$$

Here, $S_{a+b}$ denotes the set of permutations of $1,2,dotsc,a+b$.

By a result of Foata, one can also look at the pair of statistics $(maj, inv)$, and a few other combinations — these pairs of statistics will produce the same numbers.

Now, according to the OEIS entry A090806, it is proved by Garsia-Gessel, that

$$

sum_{a,b} t_{a,b} q^a t^b = prod_{i,j geq 1} frac{1}{1-q^i t^j}. (ast)

$$

I cannot see exactly where in their paper one can deduce this.

**My attempt**

I have tried to prove this myself (mainly by resorting to RSK, the Cauchy identity,

and some symmetric function identities).

This leads to the following (which appears in Stanley’s EC2):

begin{equation}

sum_{n geq 0} frac{z^n}{(1-q)^n(n)_q!(1-t)^n (n)_t!} sum_{pi in S_n} t^{maj(pi)} q^{maj(pi^{-1})}

=

prod_{i,j geq 0} frac{1}{1-z q^i t^j}.

end{equation}

where $(n)_q! := (1)_q (2)_q dotsm (n)_q$, and $(n)_q = 1+q+q^2+dotsb + q^{n-1}$.

However, I do not see some short way to deduce the above generating function from this.

**Question:** Is there some alternative (more recent?) reference where $(ast)$ is

stated and easily referenced? Alternatively, someone who can see exactly where in the paper obtains $(ast)$?

*Garsia, A. M.; Gessel, I.*, **Permutation statistics and partitions**, Adv. Math. 31, 288-305 (1979). ZBL0431.05007.