2 problems

Let $B_p(n,k)$ be the number of binary words of length $n$ with exactly $k$ occurrences of a binary word $p$ of length $\ell$ having $r\geq2$ runs. The degree of $B_p(n,k)$ is boun…

Let $B_p^{(m)}(n,k)$ denote the number of $m$-ary words of length $n$ having exactly $k$ occurrences of an $m$-ary word $p$ of length $ell$ with $r\geq 2$ runs. Let…