TITLE: A convolution-type identity for the "major index"
QUESTION [6 upvotes]: 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 $\frak{S}_n$?

REPLY [6 votes]: 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.
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.