that short blocks have length equal or smaller to $4\log_2 n$.

Permutations of blocks for good Lyndon words {#permutation}
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In the previous section, we have cut any good Lyndon word $w$ into blocks beginning with a run of “$a$” and ending with a run of “$b$”: the long ones and and the short ones. The long ones correspond to long runs of “$a$”, and the first long block (at the beginning of $w$) is, by definition of a Lyndon word, the smallest block for the lexicographic order. We are going to see that we can keep this first long block of $w$ at the beginning of the word and permute the other blocks, without changing the distribution on the set of good Lyndon words.

In the following, “short” and “long” refer to the type of blocks, while “small” and “large” refer to the lexicographic order on words.

Consider $w \in \mathcal G_n$.

1\. We denote by $K_n(w)$ the total number of blocks, long and short, of $w$.

2\. We denote by $(Y_i(w))_{0 \le i \le K_n(w)-1}$ the blocks of $w$ in their order of appearance along $w$. Certainly, the first block $Y_0(w)$ is the smallest block among all blocks 