I changed the thread completely so that everything is now elementary linear algebra.

A Dyck path of length $n$ is a list of positive integers $[c_1,c_2,...,c_n]$ with $c_i -1 \leq c_{i+1}$ for all $i$ and $c_i \geq 2$ for $i \neq n$ and $c_n=1$. (One can show that those sequences really correspond to the classical Dyck paths via the area sequence and the number of Dyck paths of length $n$ is $C_{n-1}$ when $C_n$ denotes the Catalan numbers)

Let $D=[c_1,c_2,...,c_n]$ be a Dyck path of length $n$. We define the Cartan matrix $C_D$ of $D$ as the $n \times n$ upper triangular matrix with entries 0 or 1 as follows: In the $i$-th row $C_D$ has entries equal to one in position $(i,i)$, $(i,i+1)$,...,$(i,i+c_i-1)$ and all other entries are zero. Define the Coxeter matrix $\phi_D$ as $-C_D^{-1} C_D^T$. Call a matrix $M$ periodic in case $M^k=id$ for some $k \geq 1$ and the minimal such $k$ is called the period of $M$ (and let the period be zero in case no such $k$ exists). Call a Dyck path Coxeter-periodic, or for short just periodic, in case its Coxeter matrix is periodic and the Coxeter-period, or just period, of the Dyck path is defined as the period of the Coxeter matrix.

Call a Dyck path bouncing in case it is of the form $[a_1+1,a_1,...,3,2,a_2+1,a_2,...,2,...,a_r+1,a_r,...,3,2,1]$.

I can prove that bouncing Dyck paths of length $n$ have period $n+1$ (as suggested by the comment of Michael Albert). It seems that the converse is also true:

Conjecture: A Dyck path of length $n$ is bouncing if and only if it has period $n+1$.

The conjecture is checked for $n \leq 9$.

Moreover there are the following natural questions:

What are the periodic Dyck paths and how can they be enumerated? What is the statistics of their period?

For $n \leq 5$ all Dyck paths are periodic. For $n \geq 6$ the sequence of non-periodic Dyck paths starts with 1,17,167,... See http://www.findstat.org/StatisticsDatabase/St001218 for the period of all Dyck paths with $n \leq 7$.

Here two examples: $[ 3, 4, 3, 3, 2, 1 ]$ is not periodic. $[2,5,4,3,3,2,1]$ is periodic with period 12. The Cartan matrix of $[2,5,4,3,3,2,1]$ is given by

\begin{bmatrix} 1 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 1 & 1 & 1 & 1 & 0 \\ 0 & 0 & 1 & 1 & 1 & 1 & 0 \\ 0 & 0 & 0 & 1 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ \end{bmatrix}

and the Coxeter matrix is given by \begin{bmatrix} 0 & 0 & -1 & 0 & 0 & 0 & 0 \\ -1 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 & -1 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & -1 & -1 & -1 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & -1 & -1 & -1 \\ \end{bmatrix}

Algebraic background: Dyck paths can be identified with Nakayama algebras with a linear quiver, which are finite dimensional algebras. The definition of Cartan and Coxeter matrices is then natural, see for example https://www.sciencedirect.com/science/article/pii/S0024379505001709 . In particular, the study of Coxeter matrices is interesting because it is a derived invariant of the algebra and has many applications. For example one can conclude that two algebras with different periods are not derived equivalent.

In case the conjecture is true, one would have the following nice equivalent characterisations of bouncing Dyck paths:

1.$D$ is bouncing (combinatorial characterisation).

2.$D$ is derived equivalent (=iterated tilted) to a hereditary algebra of Dynkin type $\mathcal{A}$ (homological characterisation).

3.The trivial extension of $D$ is a Brauer tree algebra (representation-theoretic characterisation).

4.The coxeter polynomial is equal to $\sum\limits_{k=0}^{n}{x^k}$ (polynomial characterisation).

5.The Coxeter matrix of $D$ has period $n+1$ (linear algebraic characterisation).

The equivalences of 1., 2., 3. and 4. is proven and the equivalence with 5. would follow from the conjecture in the thread.