fexc}(\sigma,c) := r.\#\big\{ i \in\{1,\dots,n\} \, : \, \sigma_i>i\text{ and } c_i=0 \big\} + \sum_{i=1}^{n} c_i.$$ We also need another definition of flag descents, which is the one originally due to Foata and Han [@foatahan]: $$\operatorname{fdes}^*(\sigma,c) := r . \operatorname{des}^* (\sigma,c) + c_1$$ where $$\operatorname{des}^*(\sigma,c) := \# \{ i \, : \, c_i < c_{i+1}, \text{ or } c_i=c_{i+1} \text{ and } \sigma_i>\sigma_{i+1} \}.$$

In particular, let us mention that the statistics $\operatorname{fexc}$ and $\operatorname{fdes}^*$ are equidistributed on $\mathfrak{S}_n^{(r)}$, see [@foatahan Theorem 1.4]. We will also give a proof that $\operatorname{fdes}$ and $\operatorname{fdes}^*$ are equidistributed in the next section.

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

Let $C_{n,k}^{(r)}(z)$ denote the right-hand side of the equation, and we use the same convention as with $B_{n,k}^{(r)}(z)$ when $k\leq0$. Let also $C_n^{(r)}(y,z) = \sum_{k=0}^{rn} C^{(r)}_{n,k}(z) y^k $, and $$C^{(r)}(x,y,z) = \sum_{n\geq 0} C_n^{(r)}(y,z) \frac{x^n}{n!}.
$$ From Proposition \[ehrsA\] and Theorem \[relAC\] in the sequel, we have $$C^{(r)}(x,y,z) = e^{(1-z)y^rx} A^{(r)}(x,y,z).$$ Comparing with Theorem \[relAB\] shows that we have $B^{(r)}(x,y,z) = C^{(r)}(x,y,z)$, which proves the theorem.

We have in fact another result, which is not trivially equivalent to the previous one:

