Martin Klazar's paper Irreducible and connected permutations (pdf) states:
We call a permutation $\pi$ of $[n] = \{1, 2, \ldots, n\}$ disconnected iff there is an interval $I \subset [n]$, $2 \leq |I| \leq n - 1$ such that its image $\pi(I)$ also is an interval. If $\pi$ is not disconnected, it is connected.
Then it continues:
For example, all three permutations of length 1 and 2 are connected, all six permutations of length 3 are disconnected, and the only connected permutations of length 4 are $(2, 4, 1, 3)$ and $(3, 1, 4, 2)$.
I'm having trouble understanding the definitions of "connected" and "disconnected".
For instance, consider the permutation $(1, 3, 2)$, which is claimed to be disconnected. It therefore should have interval $I$ satisfying $2 \leq |I| \leq n - 1 = 2$ whose image under this permutation is also an interval. From my calculations, the images of all the sequences of length $2$ are not intervals:
- $\pi([1, 2]) = (1, 3)$
- $\pi([2, 3]) = (3, 2)$
(Maybe the confusion comes from a weak understanding of what an interval is in this context. Isn't it a contiguous, increasing sequence?)
Furthermore, I am curious about the broader applications of these concepts.