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 .
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