A non-empty word of
is a Lyndon word if and only if it
is strictly smaller for the lexicographical order than any of its
proper right factors. Such a word
either is a letter or admits a
standard factorization
where
is its smallest proper right
factor. For any Lyndon word
, we explicitly compute the generating
function of Lyndon words having
as right factor of the standard
factorization, showing in this way that this function is rational. We
establish that, for the uniform distribution over the Lyndon words of
length
, the average length of the right factor
of the standard
factorization is asymptotically
.