Cyclic interval ($I_j^{(r)}$)

A cyclic interval of length $r$ is the set of $r$ consecutive positions in a cyclic ordering. For $1\le r<n$, there are $n$ distinct intervals, one ending at each position. Under a uniformly random permutation, a fixed $r$-set appears as an interval with probability $n/\left(\genfrac{}{}{0ex}{}{n}{r}\right)$. This is the counting mechanism of the Katona circle method.