Let $n \geq 3$ be fixed.
We associate to every subset $S \subseteq \{1,...,n-1 \}$ a number, which we call Cartan determinant of $S$ (see the end of this post for a representation theoretic background).
Define for $k,t \in \{1,...,n \}: m\_{k,t}:= t-max(0,t-k)=min(k,t)$ (so $m\_{k,t}=t$ for $k>t$ and $m\_{k,t}=k$ for $t \geq k$).


Write $S= \{s\_1,...,s\_{r-1} \}$ with the $s\_i$ in increasing order and $r-1$ the cardinality $|S|$ of $S$ and set $s\_r:=n$.


Let $a\_{i,j}:=m\_{s\_i,s\_j}=min(s\_i,s\_j)$ for $i,j \in \{1,...,r\}$.
Let $A\_S$ be the matrix with entries $a\_{i,j}$, called the Cartan matrix of $S$ and let $d\_S$ be defined as the determinant of $A\_S$, called the Cartan matrix of $S$.


We set $A\_{\emptyset}=n$ and thus $d\_{\emptyset}=n$.



> 
> Question 1: Is there an explicit formula for $d\_S$?
> 
> 
> Question 2: I can prove that $d\_S$ is non-zero. Is it more general true that $d\_S >0$?
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Now call $S$ great in case the following holds.


-In case $n$ is even, $|S|$ is odd,$min(S)$ is odd and $max(S)$ is odd and all differences $s\_i-s\_{i-1}$ are odd for $i=2,...,r-1$.


-In case $n$ is odd, $|S|$ is even, $min(S)$ is odd, $max(S)$ is even and all differences $s\_i-s\_{i-1}$ are odd for $i=2,...,r-1$.


-$\emptyset$ is great if and only if $n$ is odd.



> 
> Question 3: Is it true that $d\_S$ is odd if and only if $S$ is great? Are the great subsets $S \in \{1,...,n-1 \}$ enumerated by the Fibonacci numbers $F\_n$?
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Question 2 and 3 have a positive answer for $n \leq 14$ and thus there is good evidence that it is true and something nice is going on. I can prove (looking at entries mod 2) that if $S$ is great, then $d\_S$ is odd, but Im not sure about the converse. The result being so nice, I would think that there is a short proof that avoids heavy computations.


Now here come some more weird observations that might be much harder to show.



> 
> Question 4:
> Call a subset $S$ prime in case $d\_S$ is prime. When is $S$ prime?
> For $n \leq 20$ it was true that the number of prime $S$ is given by $\sum\limits\_{k=1}^{m}{\pi(k)}$, where $\pi(k)$ is the prime counting function. See <https://oeis.org/A046992> . Is this true in general?
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Some other findings that occured in the oeis:



> 
> The number of subsets $S$ with $d\_s=1 mod 3$ seems to be given by the seqeunce <https://oeis.org/A075111> , related to the tribonacci numbers. Is this true?
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> The number of subsets $S$ with $d\_S=1 mod 4$: <https://oeis.org/A005252> .
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> With $d\_S=3 mod 4$: <https://oeis.org/A024490>.
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Note that all the 3 previous sequences are related to the Fibonacci sequence, which might indicate that this all has a secret connection to Fibonacci combinatorics (which I can not see at the moment).



> 
> The number of $S$ with $d\_S$ such that $Gcd(d\_S,n)=1$ seem to be given by <https://oeis.org/A100347>.
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Representation theoretic background.


Let $A=A\_n=K[x]/(x^n)$. Then there is a unique indecomposable $A$-module $M\_i$ of dimension $i$ for $i=1,...,n$.


The subsets $S \subseteq \{1,...,n-1 \}$ are in bijection with the (faithful) module $M\_S := A \oplus \bigoplus\limits\_{i \in S}^{}{M\_i}$. Then $d\_S$ is the Cartan determinant of the algebra $End\_A(M\_S)$.



Cartan determinants of subsets
If $s\_1<s\_2<\ldots<s\_{r-1}<s\_r=n$, the matrix $A\_S$ is the matrix of the quadratic form
$$
s\_1(x\_1+\ldots+x\_r)^2+(s\_2-s\_1)(x\_2+\ldots+x\_r)^2+\\(s\_3-s\_2)(x\_3+\dots+x\_r)^2+\ldots+(s\_r-s\_{r-1}) x\_r^2.
$$
This quadratic form is obviously positive-definite which already implies $d\_S>0$. Next, we get $$(A\_Sx,x)=(D\_STx,Tx)\Rightarrow A\_S=T^tD\_ST,$$
where $D\_S$ is diagonal with elements $s\_1,s\_2-s\_1,\ldots,s\_r-s\_{r-1}$ and $T$ is the triangular map $x\to (x\_1+\ldots+x\_r,x\_2+\ldots+x\_r,\ldots,x\_r)$. Thus
$$
\det A\_S=\det D\_s=s\_1(s\_2-s\_1)(s\_3-s\_2)\ldots (s\_r-s\_{r-1})
$$
which answers other questions. 


