Abstract
Partial words are sequences over a finite alphabet that may contain wildcard symbols, called holes, which match, or are compatible with, all letters in the alphabet ((full) words are just partial words without holes). The subword complexity function of a partial word w over a finite alphabet A assigns to each positive integer, n, the number, pw(n), of distinct full words over A that are compatible with factors of length n of w. In this paper, with the help of our so-called hole functions, we construct infinite partial words w such that pw(n)=Θ(nα) for any real number α>1. In addition, these partial words have the property that there exist infinitely many non-negative integers m satisfying pw(m+1)−pw(m)≥mα. Combining these results with earlier ones on full words, we show that this represents a class of subword complexity functions not achievable by full words. We also construct infinite partial words with intermediate subword complexity, that is, between polynomial and exponential.
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