For $n \geq 3$ and $r \geq 3$ let $C_{n,r}=(c_{i,j})$ denote the $n \times n$-matrix where $c_{i,j}=1$ for $j=i,\dots,i+r-1$ (we only do this until $i+r-1>n$). So for example for $n=7$ and $r=3$ we obtain the matrix

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

Define $M_{n,r}:=- C_{n,r}^{-1} C_{n,r}^T$.

Recall that a matrix $M$ is called periodic in case $M^q$ is the identity matrix for some $q \geq 1$. The smallest such $q$ is called the period in case it exists.

Question 1: Given $r \geq 3$. For which $n$ are the $M_{n,r}$ periodic and what is the period in case they are periodic? What is the maximal finite period for a given $n$ or at least a good bound?

(a more general question was asked in What are the periodic Dyck paths? )

Background: The matrices $M_{n,r}$ are the Coxeter matrices of linear Nakayama algebras of the form $A_n/J^r$, where $A_n$ is the hereditary Nakayama algebra with $n$ simples and $J$ its Jacobson radical. The periods are derived invariants of those algebras. It seems to be an open problem to determine the periods, see for example page 10 in https://www.math.uni-bielefeld.de/icra2012/presentations/icra2012_lenzing.pdf for a large table and section 6 for some background of this problem https://www.sciencedirect.com/science/article/pii/S0001870813000182 .

More generally one can define the Coxeter matrix of linear Nakayama algebras as in What are the periodic Dyck paths? .

Question 2: What is the maximal period of a Coxeter matrix of a linear Nakayama algebra?

for $n=3,\dots,9$ the sequence starts with 4,6,8,12,18,30,16.