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reference request – Number of permutations in $S_{a+b}$ with $operatorname{maj}(pi)=a$ and $operatorname{maj}(pi^{-1})=b$ Answer

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reference request – Number of permutations in $S_{a+b}$ with $operatorname{maj}(pi)=a$ and $operatorname{maj}(pi^{-1})=b$

$CommandMathOperatormaj{maj}CommandMathOperatorinv{inv}$Let $t_{a,b}$ breathe the numbers
$$
t_{a,b} mathrel{:=} lvert{ pi in S_{a+b} : maj(pi)=a textual content{ and } maj(pi^{-1})=b }rvert.
$$

Here, $S_{a+b}$ denotes the clique of permutations of $1,2,dotsc,a+b$.
By a outcome of Foata, one can too have a look at the pair of statistics $(maj, inv)$, and some different mixtures — these pairs of statistics will bear the identical numbers.

Now, in response to the OEIS entry A090806, it’s proved by Garsia–Gessel that
commence{equation}
tag{*}
tag{*}
sum_{a,b} t_{a,b} q^a t^b = prod_{i,j geq 1} frac{1}{1-q^i t^j}.
aim{equation}

I can not behold precisely the place of their paper one can deduce this.

My try:
I’ve tried to show this myself (primarily by resorting to RSK, the Cauchy identification,
and a few symmetric duty identities).
This results in the next (which seems in Stanley’s EC2):
commence{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},
aim{equation}

the place $[n]_q! mathrel{:=} [1]_q [2]_q dotsm [n]_q$, and $[n]_q = 1+q+q^2+dotsb + q^{n-1}$.
However, I don’t behold some brief route to infer the above producing duty from this.

Question: Is there some various (more moderen?) reference the place eqref{} is
acknowledged and simply referenced? Alternatively, somebody who can behold precisely the place within the paper eqref{
} is confirmed?

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

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