Let $L$ be a finite distributive lattice with $n$ elements. Let $C=(c_{x,y})$ be the $n \times n$ matrix with entry 1 in case $x \leq y$ and 0 else. The Coxeter matrix of $L$ is defined as the matrix $G_L=-C^{-1}C^T$ (this is the matrix of the Auslander-Reiten translation acting on the Grothendieck group of the derived category of the poset).

I noted that for $n \leq 10$ it was always true that the permanent of $G_L$ is either 1 or -1. I was able to prove it only for some small cases such as boolean algebras and some random examples.

Question 1: Is this true in general?

Question 2: Does one have a nice order theoretic characterisation when it is 1 or -1 in case question 1 is true?

Question 3: Let $L_n$ be the set of distributive lattices with n elements. Is the sum $|\sum_{L \in L_n}^{}{Perm(G_L)}|$ bounded for $n \rightarrow \infty$? For $n \leq 10$ it was at most 2.

It is also interesting to note that for arbitrary finite lattices it seems that the permanent of $G_L$ can be arbitrary large.

My knowledge of permanents is close to zero so Im sorry in case this question is not suitable for MO.