\[theobnr2\] $$B^{(r)}_{n,k}(z) = 
   \sum_{ \substack{ (\sigma,c) \in \mathfrak{S}^{(r)}_n \\ \operatorname{fexc}(\sigma,c) = rn-k   }}  z^{ \lceil \operatorname{fdes}(\sigma,c) /r \rceil }.$$

This is a consequence of the previous theorem, together with the bijection in Section \[sec:bij\].

In view of the previous two theorems, one can ask whether the pairs $(\operatorname{fexc},\operatorname{fdes})$ and $(\operatorname{fexc},\operatorname{fdes}^*)$ are equidistributed. This is however not the case.

Chromatic descents {#sec:chro}
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For a colored permutation $(\sigma,c)$ we define its [*chromatic descent number*]{} as $$\operatorname{cdes}(\sigma,c) := \operatorname{des}(\sigma) + \sum_{i=1}^n c_i.$$

We show that it is equidistributed with the flag descent number, via a bijection $\alpha$. Let $(\sigma,c)$ be a colored permutation. Let $\alpha(\sigma,c) = (\sigma,c')$ where $$c'_i = \sum_{j=1}^i  c_j + \operatorname{des}(\sigma_1,\dots,\sigma_i) \mod r.$$

$\operatorname{fdes}(\sigma,c') = \operatorname{cdes}(\sigma,c)$.

Let $w_k = \sum_{j=1}^k c_j + \operatorname{des}(\sigma_1,\dots,\sigma_k)$ for $k=1,\dots,n$ so that $c'_i = w_i \mod r$. Clearly, $w_1,\dots,w_n$ is a nondecreasing sequence and $w_n=\operatorname{cdes}(\sigma,c)$. We can write $w_n=qr+c_n'$ for a unique $q$. This integer $q$ counts the number of positive multiples of $r$ that are smaller than $w_n$. Using the fact that $w_k-w_{k-1}\leq r$ and $w_1<r$, we have: $$q = \#\{ i \, : \,  \exists k,   w_{i-1} < kr \leq w_i   \}.$$ To count the cardinality of this set, we distinguish two 