  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.
