For a permutation $\pi\in\frak{S}\_n$, define the number of descents of $\pi$ as $$\text{des}(\pi)=\vert\{i: \pi(i)>\pi(i+1)\}\vert.$$
The following is a well-known (and interesting) identity:
$$\binom{k\ell+n-\text{des$(\pi)$}-1}n=\sum\_{\sigma\tau=\pi}
\binom{k+n-\text{des$(\sigma)$}-1}n\binom{\ell+n-\text{des$(\tau)$}-1}n.$$
This motivated me to ask:



> 
> **QUESTION.** Is there a similar identity for the **major index**? What about other "statistic" on $\frak{S}\_n$?
> 
> 
> 



A convolution-type identity for the "major index"
Richard's identity $(\*)$ can be found as Theorem 11 (though not stated exactly this way) in my paper *Multipartite 
P-partitions and inner products of skew Schur functions*, Combinatorics and algebra (Boulder, Colo., 1983), 289–317,Contemp. Math., 34, Amer. Math. Soc., Providence, RI, 1984. (It's actually $\tau\sigma=\pi$ rather than $\sigma\tau=\pi$.) You can find this paper at [http://people.brandeis.edu/~gessel/homepage/papers/multipartite.pdf](http://people.brandeis.edu/%7Egessel/homepage/papers/multipartite.pdf).


I don't think that setting $x\_i=q^{i-1}$ and $y\_j=t^{j-1}$ works except when $\pi$ is an identity permutation or the reverse of an identity permutation when $F\_{D(\pi)}$ is symmetric.


However we can get a $q$-analogue if we keep the descent number in addition to the major index, thought it's not as simple as one might like: we take $x\_i = q^{i-1}$ for $i=1,2,\dots, M$ with $x\_i=0$ for $i>M$ and $y\_j= q^{m(j-1)}$ for $j=1,2,\dots, N$ (where $N$ may be $\infty$). See Section 4 of T. Kyle Petersen,
*Cyclic descents and P-partitions*, Journal of Algebraic Combinatorics 22 (2005) 343-375, <https://arxiv.org/abs/math/0405479>.


Some related formulas can be found in D. Krob, B. Leclerc, and J.-Y.Thibon, *Noncommutative symmetric functions. II. Transformations of alphabets*,
Internat. J. Algebra Comput. 7 (1997), no. 2, 181–264.


