Compare with https://mathoverflow.net/questions/279749/permanents-of-cnakayama-algebras-ii . I made this new thread because this conjecture is much nicer and you dont have to read many assumptions before the conjecture, since no restriction on the quiver or selfinjectivity is needed and also no equivalence classes are needed to state the conjecture. 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 ] ]