The famous Cartan determinanet conjecture states that the Cartan determinant (that is the Determinant of the Cartan matrix of the algebra) of a finite dimensional algebra is equal to one in case the algebra has finite global dimension.
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.
Im not very experienced with permanents, so maybe this is an easy consequence of the Cartan determinant conjecture or maybe there exits a simple counterexample as I only looked at very special classes of algebras.