Subsequence frequency in binary words

The numbers we study in this paper are of the form Bn,p(k), which is the number of binary words of length n that contain the word p (as a subsequence, not necessarily consecutive in the word) exactly k times. Our motivation comes from the analogous study of pattern containment in permutations. In our first set of results, we obtain explicit expressions for Bn,p(k) for small values of k. We then focus on words p with at most 3 runs and study the maximum number of occurrences of p a word of length n can have. We also study the internal zeros in the sequence (Bn,p(k))k≥0 for fixed n and discuss the unimodality and log-concavity of such sequences.