length of the standard right factor. We can distinguish two cases:

-   If the word obtained from the Lyndon word by deleting the first “$a$” is still a Lyndon word, then the length of the standard right factor is equal to $n-1 \simeq n$, and this happens with a probability close to $1/2$ (this probability is replaced by $1/q$ in the case of an alphabet with $q$ letters);

-   Otherwise, the standard right factor begins by the second longest run of “$a$”. In this case, the length of the standard right factor is equal to $n$ minus the distance $D_{n}$, and is then approximately uniformly distributed on $[0..n]$.

To prove that $D_{n}$ is approximately uniformly distributed on $[0..n]$, we cut a random word with length $n$ into distinct “long blocks” with length of order $\log_2 n$ (section \[building\]), in such a way that the long runs of “$a$” are at the beginnings of the long blocks. Then we prove that the uniform distribution on Lyndon words is invariant under uniform permutation of these blocks (section \[permutation\]). Thus the positions of the two smallest (for the lexicographic order) long blocks are approximately uniformly distributed among all the possible positions of the 