I was reading the proof of the closed-form formula of Eulerian numbers ($A(n,k)$) from Bona's book. I had a doubt in his classification of "removable fences" and how they eliminate the undesirable permutations. 
When we are counting the total number of desired permutations, we see that 2 things might happen which lead to undesirable permutations (| is a fence and signifies the separation of the compartments/concessions):
- Empty compartments/concessions (eg: 234||1)
- Neighboring compartments/concessions with no descent in between (eg: 12|3)
Bona classifies a fence $f$ as removable if it satisfies these 2 conditions:
- If we remove $f$, we still have a permutation whose entries increase between any two consecutive fences, and
- the fence $f$ is not immediately followed by another fence
He then says that any permutation that doesn't contain any removable fence has $k$ ascending runs. I do not understand why this definition of removable works (if possible, please explain through examples) and why this eliminates undesirable permutations. For instance, say we are counting $A(6,3)$, it still counts permutations of type 124||356, which isn't a desirable permutation as it only has 2 ascending runs.