This is a continuation of Classification of algebras of finite global dimension via determinants of certain 0-1-matrices but this time with a concrete conjecture and using the simplification suggested by user esg to make the problem more compact.

Let $n \geq 4$ and $w \in \{3,4,...,n-1 \}$. Let $Z$ be the matrix of the cyclic shift (the companion matrix of $X^n-1$), and for non-zero $\mathbf{v}\in \{0,1\}^n$ let $\mathrm{diag}(\mathbf{v})$ be the diagonal matrix with $\mathbf{v}$ on the diagonal, and $M_\mathbf{v}:=I + Z+ \ldots + Z^{w-1}-\mathrm{diag}(\mathbf{v})$.
For fixed $n$, call the tuple $(w,v)$ perfect in case $\det(M_\mathbf{v})=(-1)^{(w-1)(n-1)}$. (here I used the reformulation obtained by user esg for my problem. I refer to the previous thread for a motivation. Roughly stated, perfect pairs correspond to certain algebras with finite global dimension. )

Define $G_n := \{ w \in \{3,4,...,n-1\} | $there exists a nonzero $\mathbf{v}\in \{0,1\}^n$ with $(w,v)$ perfect $\}$.

It is best to picture the $v$ as two-colored necklaces (with colours corresponding to 1 and 0), so a cyclic shift just means rotating the necklace.

It is an interesting question what the set $G_n$ is explicitly but my first guess was wrong and it seems that $G_n$ is complicated to describe for large $n$.

But here are two conjectures that would be nice in case they are true:

Conjecture 1: Maximum($G_n$)=$\frac{n+2}{2}$ in case $n$ is even and Maximum($G_n$)=$\frac{n+1}{2}$ in case $n$ is odd.

Conjecture 2: a) For $n$ even the number of perfect tuples $(w,v)$ with $w=(n+2)/2$ is equal to $\frac{3^{n/2-1}+1}{2}$ when we identify two tuples $(w,v_1)$ and $(w,v_2)$ when $v_1$ is a cyclic shift of $v_2$.

b) For $n$ odd the number of perfect tuples $(w,v)$ with $w=(n+1)/2$ is equal to $n-1$ when we identify two tuples $(w,v_1)$ and $(w,v_2)$ when $v_1$ is a cyclic shift of $v_2$.

The conjectures are tested with the computer for $n \leq 20$.

Here two examples:

For $n=13$ and $w=7$, the $v$ up to cyclic shift with $(w,v)$ perfect are:

[ [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 ],

[ 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1 ],

[ 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1 ],

[ 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1 ],

[ 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1 ],

[ 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1 ],

[ 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1 ],

[ 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1 ],

[ 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1 ],

[ 0, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1 ],

[ 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1 ],

[ 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ] ]

For $n=8$ and $w=5$, the perfect $(w,v)$ up to cyclic shifts are:

[ [ 0, 0, 0, 1, 0, 0, 0, 1 ],

[ 0, 0, 0, 1, 0, 0, 1, 1 ],

[ 0, 0, 0, 1, 0, 1, 0, 1 ],

[ 0, 0, 0, 1, 0, 1, 1, 1 ],

[ 0, 0, 0, 1, 1, 0, 0, 1 ],

[ 0, 0, 0, 1, 1, 0, 1, 1 ],

[ 0, 0, 0, 1, 1, 1, 0, 1 ],

[ 0, 0, 0, 1, 1, 1, 1, 1 ],

[ 0, 0, 1, 0, 0, 1, 1, 1 ],

[ 0, 0, 1, 0, 1, 0, 1, 1 ],

[ 0, 0, 1, 0, 1, 1, 1, 1 ],

[ 0, 0, 1, 1, 0, 1, 0, 1 ],

[ 0, 0, 1, 1, 1, 1, 0, 1 ],

[ 0, 1, 0, 1, 1, 0, 1, 1 ] ]

edit: It might be also a good idea to think about conjecture 1 in terms of representation theory/homological algebra. Here is the non-elementary formulation of conjecture 1:

Let $A$ be a selfinjective (connected) Nakayama algebra with $n$ simple modules and Loewy length $w$ with $(n+2)/2 < w<n$. Then a generator $M$ with every non-projective indecomposable summand being simple has the property that $End_A(M)$ has infinite global dimension.

(equivalently, one can look at generators $M$ with every non-projective indecomposable summand being a radical of an indecomposable projective module).