cases. If $w_{i-1}-w_i=r$, it means that $c_i=r-1$ and $\sigma_{i-1}>\sigma_i$. From the definition of the bijection, this is equivalent to $c'_i=c'_{i+1}$ and $\sigma_{i-1}>\sigma_i$. Otherwise, $w_{i-1}-w_i<r$. We can see that this case is equivalent to $c_{i-1}>c_{i}$. Hence, we obtain $q=\operatorname{des}(\sigma,c')$, and $w_n=\operatorname{fdes}(\sigma,c')$.

It is also possible to define a bijection $\alpha^*$ by $\alpha^*(\sigma,c)=(\sigma,c'')$ where $$c''_i = \sum_{j=i}^n c_j + \operatorname{des}(\sigma_i,\dots,\sigma_n) \mod r.$$ As in the case of the previous proposition, we can prove $\operatorname{fdes}^*(\sigma,c'') = \operatorname{cdes}(\sigma,c)$. In particular, it follows that $\operatorname{fdes}$ and $\operatorname{fdes}^*$ are equidistributed.

The bijection $\alpha$ only changes the colors $c_i$, and not the permutation $\sigma$, so we have: $$\sum_{(\sigma,c) \in \mathfrak{S}_n^{(r)} } y^{ \operatorname{fdes}(\sigma,c) } z^{ \operatorname{des}(\sigma^{-1}) } 
  = \sum_{(\sigma,c) \in \mathfrak{S}_n^{(r)} } y^{ \operatorname{cdes}(\sigma,c) } z^{ \operatorname{des}(\sigma^{-1}) } .$$ But the right-hand side clearly can be factorized, so that with the notation $$A^{(r)}_n(y,z) = \sum_{ k=0 }^{ rn } y^k A^{(r)}_{n,k}(z),$$ we have: $$\label{relara1}
  A^{(r)}_n(y,z) = \left( \frac{1-y^r}{1-y} \right)^n A^{(1)}_n(y,z).$$

A formula for the case $r=1$ is given in the proposition below. This is in fact a particular case of a result of Garsia and Gessel [@garsia Theorem 2.3], but we also include a short proof based on the Robinson-Schensted correspondence.

For $r=1$, we have: $$\frac{A^{(1)}_n(y,z)}{(1-y)^{n+1}(1-z)^{n+1}} = 