group structure of colored permutations. Still, note that flag descents are indeed related with it [@bagno1].

To define our generalization of Stanley’s bijection, let $ (a_i)_{1\leq i \leq n} \in [0,r)^n$, and let $a_0=0$. Then the map $\phi( (a_i)_{1\leq i \leq n} ) = (b_i)_{1\leq i \leq n} $ is defined as follows: $$\label{defphi}
  b_i = \begin{cases}
          a_i-a_{i-1} & \text{ if } a_{i-1}\leq a_i, \\
          a_i-a_{i-1}+r & \text{ if } a_{i-1}> a_i.
        \end{cases}$$ From this definition we get: $$\label{inversephi}
  a_i = \sum_{j=1}^i b_j  \mod r,$$ where the modulo means that we take the unique representative in $[0,r)$. In fact, it is elementary to check that Equations  and  define two inverse bijections from $[0,r)^n$ to itself.

If $v\in [0,r)^n$, we define $\operatorname{fdes}(v)=r.\operatorname{des}(v) +  v_n $. If $1 \leq k \leq rn $, let $$\mathcal{F}^{(r)}_{n,k} := \{ v \in [0,r)^n \, : \, k-1 \leq \operatorname{fdes}(v) < k \},$$ and $$\mathcal{A}^{(r)}_{n,k} := \{ v \in [0,r)^n \, : \, k-1 \leq \sum v_i < k \}.$$ For each colored permutation $(\sigma,c)$ we define the translated simplex: $$\mathcal{T}_{(\sigma,c)} := c + \mathcal{T}_{\sigma}.$$ Also, the [*colored standardization*]{} of $v\in [0,r)^n$ is $\operatorname{cstd}(v) = (\sigma,c) \in \mathfrak{S}^{(r)}_n$ where $c_i = \lfloor v_i \rfloor $ and 