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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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