The number of cycles in the breakpoint graph of a permutation. The breakpoint graph of a permutation $\pi_1,\dots,\pi_n$ is the directed, bicoloured graph with vertices $0,\dots,n$, a grey edge from $i$ to $i+1$ and a black edge from $\pi_i$ to $\pi_{i-1}$ for $0\leq i\leq n$, all indices taken modulo $n+1$. This graph decomposes into alternating cycles, which this statistic counts. The distribution of this statistic on permutations of $n-1$ is, according to [cor.1, 5] and [eq.6, 6], given by $$\frac{1}{n(n+1)}((q+n)_{n+1}-(q)_{n+1}),$$ where $(x)_n=x(x-1)\dots(x-n+1)$.