Subsequence frequency in binary words

The numbers we study in this paper are of the form $B_{n, p}(k)$, which is the number of binary words of length $n$ that contain the word $p$ (as a subsequence) 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 $B_{n, p}(k)$ 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 $(B_{n, p}(k))_{k \geq 0}$ for fixed $n$ and discuss the unimodality and log-concavity of such sequences.