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.

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 ] ]