The standard factorization of Lyndon words: an average point of view Original Research Article

A non-empty word w is a Lyndon word if and only if it is strictly smaller for the lexicographical order than any of its proper suffixes. Such a word w is either a letter or admits a standard factorization uvuv where vv is its smallest proper suffix. For any Lyndon word vv, we show that the set of Lyndon words having vv as right factor of the standard factorization is regular and compute explicitly the associated generating function. Next, considering the Lyndon words of length n over a two-letter alphabet, we establish that, for the uniform distribution, the average length of the right factor vv of the standard factorization is asymptotically 3n/43n/4.