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

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.