\,:\,  k \leq i <m \text{ and } B_i > B_{i+1}  \big\}$$ to $\lceil \operatorname{fdes}(\sigma,c) /r \rceil$ (the 1 come from the term $c_n$ in the definition of $\operatorname{fdes}$ since $B_m$ is a nonzero colored block), and the suffix $B_k B_ m \dots B_{k+1}$ of $I(\sigma,c)$ contributes by $$1 + \#\big\{ i \,:\,  m > i \geq k+1 \text{ and } B_{i+1} < B_i  \big\}$$ to $\lceil \operatorname{fdes}^*(I(\sigma,c)) /r \rceil$ (the 1 come from the fact that $B_k<B_m$). The two numbers are easily seen to be equal.

Checking the respective definitions of $\operatorname{fdes}$ and $\operatorname{fdes}^*$, we can see that what we have counted proves the proposition.

Formulas for the Ehrhart polynomials {#sec:ehrform}
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The Ehrhart polynomial of $\mathcal{A}^{(r)}_{n,k}$ is: $$\begin{aligned}
      \sum_{j=0}^{\lfloor (k-1)/r \rfloor} (-1)^{j+1}  \tbinom nj \tbinom{n-rtj+kt-t-1}{n} -  
                      \sum_{j=0}^{\lfloor k/r \rfloor  } (-1)^{j+1}  \tbinom nj \tbinom{n-rtj+kt-1}{n}.\end{aligned}$$

Let ${\rm CT}_q$ denote the operator that gives the constant term of a Laurent series in $q$. We have: $$\# \big( \mathbb{Z}^n \cap t\mathcal{A}^{(r)}_{n,k} \big)  
                   = \# \big\{ v \in \{0,1,\dots,rt-1\}^n  \, : \,  kt-t \leq \sum v_i < kt \big\}$$

$$=  {\rm CT}_q \bigg(  [rt]_q^n  ( [kt]_{q^{-1}} - [kt-t]_{q^{-1}} )  \bigg)$$

$$= {\rm CT}_q \bigg(  \frac{ (1-q^{rt})^n  ( q^{-kt+t-1} - q^{-kt-1 } )   