Higher Auslander algebras of finite representation type

Let $\mathrm{\Lambda }$ be an $n$-Auslander algebra with global dimension $n+1$. In this paper, we prove that $\mathrm{\Lambda }$ is representation-finite if and only if the number of non-isomorphic indecomposable $\mathrm{\Lambda }$-modules with projective dimension $n+1$ is finite. As an application, we classify the representation-finite higher Auslander algebras of linearly oriented type $\mathbb{A}$ in the sense of Iyama and calculate the number of non-isomorphic indecomposable modules over these algebras.