Realising perfect derived categories of Auslander algebras of type A as Fukaya-Seidel categories

We prove that the Fukaya-Seidel categories of a certain family of Lefschetz fibrations on $\mathbb{C}^2$ are equivalent to the perfect derived categories of Auslander algebras of Dynkin type $\mathbb{A}$. We give an explicit equivalence between these categories and the partially wrapped Fukaya categories considered by Dyckerhoff-Jasso-Lekili. We provide a complete description of the Milnor fibre of such fibrations.Comment: 51 pages, 19 figures. To appear in Journal of Symplectic Geometry (2023). Revised after referee's comments. Updated Section 3.2.2 to also give an algebraic proof of the derived equivalence of Auslander algebras of type A and $\tilde{\Gamma}_n$. Local changes throughou