Squares in Binary Partial Words

In this paper, we investigate the number of positions that do not start a square, the number of square occurrences, and the number of distinct squares in binary partial words. Letting σh(n) be the maximum number of positions not starting a square for binary partial words with h holes of length n, we show that limσh(n)/n = 15/31 provided the limit of h/n is zero. Letting γh(n) be the minimum number of square occurrences in a binary partial word of length n with h holes, we show, under some condition on h, that lim γh(n)/n = 103/187. Both limits turn out to match with the known limits for binary full words. We also bound the difference between the maximum number of distinct squares in a binary partial word and that of a binary full word by (2−1)(n+2), where n is the length and h is the number of holes. This allows us to find a simple proof of the known 3n upper bound in a one-hole binary partial word using the completions of such a partial word.