into the subsets $$\mathcal{Y}_{U} = \{ T\in\mathcal{Y}_\lambda \, : \, \operatorname{std}(T)=U \}$$ where $U\in SYT(\lambda)$. Now, the previous proposition is a consequence of the following:

$ E^*( \mathcal{Y}_U , z ) = z^{\operatorname{des}(U)+1}$.

This is essentially the same as Proposition \[ehrtsigma\].

The generalization of Stanley’s bijection {#sec:stanbij}
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In this section we adapt Stanley’s bijection from [@stanley] to the case of the half-open hypercube $[0,r)^n$ (for some integer $r>0$), and $r$-colored permutations. Note that a similar generalization was given by Steingrímsson in [@steingrimsson Section 4.4].

The set of $r$-[*colored permutations*]{} $\mathfrak{S}_n^{(r)}$ is the set of pairs $(\sigma,c)$ where $\sigma\in\mathfrak{S}_n$, $c=(c_i)_{1\leq i \leq n}$, and $c_i \in \{ 0,1, \dots,r-1 \}$ for all $i$. We define the [*descent number*]{} of a colored permutation as: $${\rm des}(\sigma,c) := \# \{ i \, : \, c_i>c_{i+1}, \text{ or } c_i=c_{i+1} \text{ and } \sigma_i>\sigma_{i+1} \},$$ and we define the [*flag descent number*]{} [@bagno1; @foatahan] as: $$\operatorname{fdes}(\sigma,c) :=  r. {\rm des}(\sigma,c) + c_n.$$ The [*flag Eulerian numbers*]{} are defined by $A^{(r)}_{0,0}:=1$, $A^{(r)}_{n,0}:=0$ if $n>0$, and $$A_{n,k}^{(r)} := \# \big\{ \sigma \in \mathfrak{S}_n^{(r)} \, : \,  {\rm fdes}(\sigma) = k-1 \big\}$$ if $n\geq 1$ and $1\leq k \leq rn$.

We will not use here the 