For $n \geq 2$, a natural number $a \geq 2$ and a number $s$ with $1 \leq s \leq n-1$ define configurations (just a list of integers) $N_{n,a,s}=[c_0,c_1,...,c_{n-1}]=[a,a,a,...,a+1,....,a+1]$, where the list has $n$ entries (numbered from 0 to $n-1$) and $a$ appears $s$ times.

Associated to this configuration is the set $ \bigcup\limits_{i \in \mathbb{Z}/ \mathbb{Z} n} \{ i \} \times \{ 1,...,c_i \}$ of 2-tupels, whose elements are called modules.

Define the $i$-th syzygy $O_i(x,y)$ of a module $(x,y)$ as $O_0(x,y)=(x,y)$, $O_1(x,y)=(x+y,c_x-y)$ and $O_{i+1}(x,y)=O_1(O_i(x,y))$. Call the modules of the form $(i,c_i)$ projective and call a module evil in case it is of the form $(0,y)$.

Define the projective dimension of a module $(x,y)$ as $pd(x,y)= \inf \{ i \geq 0 | O_i(x,y)$ is projective $\}$.

Define the finitistic dimension of the configuration as $fd(n,a,s)= sup \{ pd(x,y) | pd(x,y) $ is finite $\}$.

Define the dominant dimension $dd(n,s,a)$ of the algebra as $inf \{ i \geq 1 | O_i(s,a)$ is evil $\}$.

I can show by using rather deep theorems from homological algebra that $dd(n,s,a)=fd(n,s,a)$ for any configuration.

Is there a direct elementary proof of this combinatorial result? (One can picture such algebras/configuations as certain infinite but periodic Dyck paths)

Bonus question would be to find a formula for the finitistic or dominant dimension and decide when it is even or odd. I only partially succeded in doing so. See also Closed formula for some dimension.