On the combinatorics of finite words

Abstract

AbstractIn this paper we consider a combinatorial method for the analysis of finite words recently introduced in Colosimo and de Luca (Special factors in biological strings, preprint 97/42, Dipt. Matematica, Univ. di Roma) for the study of biological macromolecules. The method is based on the analysis of (right) special factors of a given word. A factor u of a word w is special if there exist at least two occurrences of the factor u in w followed on the right by two distinct letters. We show that in the combinatorics of finite words two parameters play an essential role. The first, denoted by R, represents the minimal integer such that there do not exist special factors of w of length R. The second, that we denote by K, is the minimal length of a factor of w which cannot be extended on the right in a factor of w. Some new results are proved. In particular, a new characterization in terms of special factors and of R and K is given for the set PER of all words w having two periods p and q which are coprimes and such that ¦w¦ = p + q − 2