of $w$.

3\. Let $j_0(w)$ be the index of the second smallest block of $w$.

4\. We denote by $\mathfrak{S}_{K_n(w)-1}$ the set of permutations of $\{1, \dots, K_n(w)-1\}$, and define $$\sigma.w=Y_0(w)Y_{\sigma(1)}(w)\dots
Y_{\sigma(K_n(w)-1)}(w),$$ for $\sigma \in
\mathfrak{S}_{K_n(w)-1}$. Obviously, $\sigma.w \in \mathcal G_n$.

5\. We define also $C(w)=\{\sigma.w, \sigma \in
\mathfrak{S}_{K_n(w)-1} \}$, the set of all words which are obtained by the all the permutations of the blocks of $w$.

The two cases of right factor exposed in the introduction can be rephrased in the following manner: either the standard right factor is obtained by deleting the first “$a$”, or it begins by the second smallest block of $w$, $Y_{j_0(w)}$.

In this section, we study the asymptotics of the position between the two smallest blocks, and we will rephrase this result in terms of standard right factor in the next section.

Our main tool is the immediate following property:

Let $w_0$ be a fixed good Lyndon word. Consider the set $\mathfrak{S}_{K_n(w_0)-1} \times
  C(w_0)$, endowed with the uniform probability. Then the random variable $$\begin{array}{rrll}
W_{w_0}: & \mathfrak{S}_{K_n(w_0)-1} \times
  C(w_0) & \longrightarrow & C(w_0) \\
& (\sigma, w) & \longmapsto & \sigma.w
\end{array}$$ follows the uniform law on $C(w_0)$.

**Proof.** It is sufficient to 