Abelian-primitive partial words

In this paper we count the number of abelian-primitive partial words of a given length over a given alphabet size, which are partial words that are not abelian powers. Partial words are sequences that may have undefined positions called holes. This combinatorial problem was considered recently for full words (those without holes). It turns out that, even for the full word case, it is a nontrivial problem as opposed to the counting of the number of primitive full words, well-known to be easily derived using the Möbius function.