Squares in partial words

We investigate the number of positions that do not start a square, the number of square occurrences, and the number of distinct squares in partial words, i.e., sequences that may have undefined positions called holes. We show that the limit of the ratio of the maximum number of positions not starting a square in a binary partial word with h holes over its length n is 15/31 and the limit of the ratio of the minimum number of square occurrences in a binary partial word with h holes over its length n   is 103/187, provided the limit of h/nh/n is 0. Both limits turn out to match with the known limits for binary full words (those without holes). We prove another surprising result that the maximal proportion of defined positions that are square-free to the number of defined positions in a binary partial word with h holes of length n   is 1/2, provided the limit of h/nh/n is in the interval [1/11,1)[1/11,1). We also give a 2kh2kh tight bound on the number of rightmost occurrences of squares per position in a k-ary partial word with h holes. In addition, we provide a more detailed analysis than earlier ones for the maximum number of distinct squares in a one-hole partial word of length n over an alphabet of size k, bound that is independent of k.