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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 $v_2$.

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 $v_2$.

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).

The conjectures are tested with the computer for $n \leq 18$.

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