See https://en.wikipedia.org/wiki/Nakayama_algebra for the definition of Nakayama algebras and define the permanent of such an algebra to be the permanent of its Cartan matrix. (all algebras are assumed to be connected and finite dimensional) Conjecture: Let $X$ be the set of Nakayama algebras of finite global dimension with $n$ simple modules. Then the maximum of the permanents of algebras in X is given by $\sum\limits_{k=0}^{\infty}{\frac{k^n}{2^{k+1}}}$ and it is uniquely attained. See https://oeis.org/A000670 for the conjectured sequence.

The algebras with a line as a quiver should always have permanent equal to one. Here the algebras with permanent higher than 1 and finite global dimension with 3 simple modules (given by their Kupisch series as first entry and the permanent as second entry):

[ [ [ 2, 2, 3 ] 3 ], [ [ 2, 4, 3 ], 5 ], [ [ 3, 4, 4 ], 11 ], [ [ 3, 5, 4 ], 13 ] ]