Counting bordered and primitive words with a fixed weight

Abstract

A word $w$ is primitive if it is not a proper power of another word, and $w$ is unbordered if it has no prefix that is also a suffix of $w$. We study the number of primitive and unbordered words $w$ with a fixed weight, that is, words for which the Parikh vector of $w$ is a fixed vector. Moreover, we estimate the number of words that have a unique border.