A Generating Function for the Distribution of Runs in Binary Words

Let $N(n,r,k)$ denote the number of binary words of length $n$ that begin with $0$ and contain exactly $k$ runs (i.e., maximal subwords of identical consecutive symbols) of length $r$. We show that the generating function for the sequence $N(n,r,0)$, $n=0,1,\dots$, is $(1-x)(1-2x+x^r-x^{r+1}{)}^{-1}$ and that the generating function for $\{N(n,r,k)\}$ is $x^{kr}$ time the $k+1$ power of this. We extend to counts of words containing exactly $k$ runs of $1$s by using symmetries on the set of binary words.