from Theorem \[ogfC\], we also get: $$\mathcal{L} \big( C^{(r)}(x,y,z) , x, s \big) = 
  \frac{(1-y)(1-z)}{s-(1-z)y^r}\sum_{m\geq 0} z^m \Bigl(1-y\bigl(\frac{s-(1-z)y^r}{s-(1-z)}\bigr)^m   \Bigr)^{-1}  .$$ So we have proved $$\mathcal{L} \big( C^{(r)}(x,y,z) , x, s \big) = \mathcal{L}\big( e^{(1-z)y^rx} A^{(r)}(x,y,z) , x , s \big),$$ which completes the proof of Theorem \[relAC\].

Another combinatorial model {#sec:bij}
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We give in this section a bijective proof of $$\sum_{ (\sigma,c) \in \mathfrak{S}_n^{(r)} } y^{ {\operatorname{fexc}}(\sigma,c)  } z^{ \lceil {\operatorname{fdes}}(\sigma,c) /r \rceil }
  = \sum_{ (\sigma,c) \in \mathfrak{S}_n^{(r)} } y^{ {\operatorname{fexc}}(\sigma,c)  } z^{ \lceil {\operatorname{fdes}^*}(\sigma,c) /r \rceil },$$ by defining an involution $I$ on colored permutations such that $$y^{ {\operatorname{fexc}}(\sigma,c)  } z^{ \lceil {\operatorname{fdes}}(\sigma,c) /r \rceil } = y^{ {\operatorname{fexc}}(I(\sigma,c))  } z^{ \lceil {\operatorname{fdes}^*}(I(\sigma,c)) /r \rceil }.$$ Let $(\sigma,c) \in \mathfrak{S}^{(r)}_n$. We consider $(\sigma,c)$ as a word whose successive letters are $(\sigma_1,c_1)$, $(\sigma_2,c_2),\dots$, $(\sigma_n,c_n)$. Note that the pair $(\sigma_i,c_i)$ is considered as a letter with color $c_i$. Then, we consider the unique factorization $$(\sigma_1,c_1) \dots (\sigma_n,c_n) =  B_1 \dots B_m$$ where each block $B_i$ contains letters of the same color, and $m$ is minimal. The involution $I$ is defined by permuting the blocks, following these two conditions:

-   each zero colored block stays at the same location,

-   