Definitions

Let $\pi = (\pi_1,\ldots,\pi_n)$ denote a permutation of $\{1,\ldots,n\}$. Augment $\pi$ to a permutation on n+2 vertices by adding $\pi_0=0$ and $\pi_{n+1} = n+1$ to it. A pair $(\pi_i,\pi_{i+1})$, $0\le i\le n$ is called a gap. Gaps are classified into two types: A gap $(\pi_i,\pi_{i+1})$ is called a breakpoint of $\pi$ if $\vert\pi_i-\pi_{i+1}\vert > 1$; otherwise, it is called an adjacency of $\pi$. We denote by $b(\pi)$the number of breakpoints in $\pi$.

Recall from section 10.4.1 that a reversal, $\rho(i,j)$, on a permutation $\pi$ transforms $\pi$ to

$$\begin{eqnarray*}\pi'=\pi\cdot \rho(i,j) = (\pi_1,\ldots,\pi_{i-1},-\pi_j, - \pi_{j-1},\ldots,- \pi_i,\pi_{j+1},\ldots,\pi_n) \end{eqnarray*}$$

We say that a reversal $\rho(i,j)$ acts on the gaps $(\pi_{i-1},\pi_i)$ and $(\pi_j,\pi_{j+1})$.