Define the Cartan permanent of a finite dimensional algebra as the permanent of the Cartan matrix. 
Is the Cartan permanent of a finite dimensional algebra with finite global dimension always an odd integer?
This is true for acyclic quiver algebras and experiments suggest that it is also true for all Nakayama algebras.



Is the Cartan permanent odd for finite global dimension?
A theorem of Eilenberg asserts that the Cartan determinant of a finite-dimensional algebra of finite global dimension is equal to $\pm 1$. Hence the Cartan determinant and the Cartan permanent are both congruent to $1$ modulo $2$ in this situation. 


The result is Proposition 21 in the following reference.


* Samuel Eilenberg, [Algebras of cohomologically finite dimension](https://link.springer.com/article/10.1007%2FBF02566937?LI=true), Comment. Math. Helv. **28** (1954), no.1, 310--319.


