linear algebra - Should the formula for the inverse of a 2x2 matrix be obvious?

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

Hello pricey customer to our community We will proffer you an answer to this query reference request – Number of permutations in $S_{a+b}$ with $maj(pi)=a$ and $maj(pi^{-1})=b$ ,and the respond will breathe typical by documented info sources, We welcome you and proffer you fresh questions and solutions, Many customer are questioning in regards to the respond to this query.

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

Let $
t_{a,b}$
breathe the numbers
$$
t_{a,b} := |{ pi in S_{a+b} : mathrm{maj}(pi)=a textual content{ and } mathrm{maj}(pi^{-1})=b }|.
$$

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

Now, in line with the OEIS entry A090806, it’s 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 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 id,
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 $[n]_q! := [1]_q [2]_q dotsm [n]_q$, 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 (newer?) reference the place $(ast)$ is
acknowledged and simply referenced? Alternatively, somebody who can behold precisely the place within the paper obtains $(ast)$?

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

we’ll proffer you the answer to reference request – Number of permutations in $S_{a+b}$ with $maj(pi)=a$ and $maj(pi^{-1})=b$ query by way of our community which brings all of the solutions from a number of dependable sources.

Add comment