TITLE: Derived equivalences of Dyck paths
QUESTION [10 upvotes]: 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? 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?

REPLY [6 votes]: Question 1 has a positive answer by the answer of Gjergji Zaimi in this thread: What are the periodic Dyck paths? .
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)