Binary De Bruijn Partial Words with One Hole

In this paper, we investigate partial words, or finite sequences that may have some undefined positions called holes, of maximum subword complexity. The subword complexity function of a partial word w over a given alphabet of size k assigns to each positive integer n, the number pw(n) of distinct full words over the alphabet that are compatible with factors of length n of w. For positive integers n, h and k, we introduce the concept of a de Bruijn partial word of order n with h holes over an alphabet A of size k, as being a partial word w with h holes over A of minimal length with the property that pw(n) = k . We are concerned with the following three questions: (1) What is the length of k-ary de Bruijn partial words of order n with h holes? (2) What is an efficient method for generating such partial words? (3) How many such partial words are there?