Call two Dyck paths $D\_1$ and $D\_2$ derived equivalent in case their corresponding Nakayama algebras are derived equivalent (The Dyck path of a Nakayama algebra with a linear quiver is just the top boundary of its Auslander-Reiten quiver, so we can identify Nakayama algebras with a linear quiver with Dyck paths). 
Derived categories of Nakayama algebras appeared in the literature (see for example <https://www.sciencedirect.com/science/article/pii/S0001870813000182> ) but it seems to be wide open to classify Dyck paths with respect to derived equivalences (or?).


Call a Nakayama algebra (or the corresponding Dyck path) bouncing in case the Kupisch series is of the form $[a\_1+1,a\_1,...,3,2,a\_2+1,a\_2,...,2,...,a\_r+1,a\_r,...,3,2,1]$.


The truth of the conjecture in [What are the periodic Dyck paths?](https://mathoverflow.net/questions/303487/what-are-the-periodic-dyck-paths) would imply the following:
In case a Dyck path $D\_1$ is derived equivalent to a bouncing Dyck path $D\_2$, then also $D\_1$ is bouncing. Thus, it would give that being bouncing is a derived invariant.


Question 1: 



> 
> Do we know whether it is true that being bouncing is a derived invariant? That is:
>  In case a Nakayama algebra $D\_1$ is derived equivalent to a bouncing Nakayama algebra $D\_2$, then also $D\_1$ is bouncing?
> 
> 
> 


Question 2:



> 
> Is it known which bouncing Nakayama algebras with a fixed number of simple modules are derived equivalent to eachother? How many equivalence classes (up to derived equivalence) exist of bouncing Nakayama algebras?
> 
> 
> 



Derived equivalences of Dyck paths
Question 1 has a positive answer by the answer of Gjergji Zaimi in this thread: [What are the periodic Dyck paths?](https://mathoverflow.net/questions/303487/what-are-the-periodic-dyck-paths/311511#311511) .


Here a positive answer to question 1 and 2 using a complicated classification result that uses several other deep results from the literature (so more elementary proofs of Question 2 are welcome):


Question 1 and 2 have a positive answer by the classification of iterated tilted algebras of Dynkin type $\mathcal{A}$.
See the main theorem of <https://www.tandfonline.com/doi/abs/10.1080/00927878108822697> , which implies here that the bouncing Dyck paths correspond to Nakayama algebras with a linear quiver having only relations of length two.


(edit: I think I found an elementary answer to Question 2 using tilting modules. I might post it here soon)


