Classification silted algebras for a quiver of Dynkin type ${\mathbb{A}}_n$ via geometric models

Let $\text{Q}$ be the quiver of Dynkin type ${\mathbb{A}}_n$ with linear orientation and $A_n=k\text{Q}$. In this paper, we give a complete classification of the silted algebras of type $A_n$ by using the geometric models of gentle algebras. We show that any finite-dimensional algebra is a silted of type $A_n$ if and only if it is a tilted of type $A_n$ or a tilted algebra of type $A_m\times A_{n-m}$ for any positive integer $1\le m\le n-1$. Based on the classification, we obtain a formula for computing the number of silted algebras of type $A_n$.