each maximal sequence of nonzero colored blocks $B_j \dots B_k$ is replaced with $B_k \dots B_j$ (maximal means that $B_{j-1}$ is zero colored or $j=1$, and $B_{k+1}$ is zero colored or $k=n$).

For example, with $n=8$ and $r=3$: $$(8,1) (2,0) (7,2) (1,2) (4,1) (3,0) (5,1) (6,1)$$ is sent to $$(8,1) (2,0) (4,1) (7,2) (1,2) (3,0) (5,1) (6,1).$$

$\operatorname{fexc}(\sigma,c) = \operatorname{fexc}(I(\sigma,c))$.

This is immediate, since the letters with color $0$ are unchanged by $I$ (and the sum of the colors is also preserved).

$\lceil \operatorname{fdes}(\sigma,c) /r \rceil = \lceil \operatorname{fdes}^*(I(\sigma,c)) /r \rceil$.

We compute $\lceil \operatorname{fdes}(\sigma,c) /r \rceil$ on one side and $\lceil \operatorname{fdes}^*(I(\sigma,c)) /r \rceil$ and the other side, by examining the different contributions to each quantity.

First, each pair of letters $(\sigma_i,c_i) (\sigma_{i+1},c_{i+1})$ where $\sigma_i>\sigma_{i+1}$ inside a given block $B_j$ contribute by 1 to each side (since $c_i=c_{i+1}$ by definition of the blocks). It remains to consider the term $r\times\#\{ i \, :\, c_i > c_{i+1}  \} + c_n$ in the definition of $\operatorname{fdes}$, and the term $r\times\#\{ i \, :\, c_i < c_{i+1}  \} + c_1$ in the definition of $\operatorname{fdes}^*$.

Let us write $B_i>B_{i+1}$ or $B_i<B_{i+1}$ to mean that the color of the block $B_i$ is greater 