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 $|\sum_{L \in L_n}^{}{\mathrm{Perm}(G_L)}|$ 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 I'm sorry in case this question is not suitable for MO.

The values of this statistic for posets has been entered recently here: http://www.findstat.org/StatisticsDatabase/St001472 .