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We build a bijection between the set $\sttilt\Lambda$ of isomorphism classesof basic support $\tau$-tilting modules over the Auslander algebra $\Lambda$ of$K[x]/(x^n)$ and the symmetric group $\mathfrak{S}_{n+1}$, which is ananti-isomorphism of partially ordered sets with respect to the generation orderon $\sttilt\Lambda$ and the left order on $\mathfrak{S}_{n+1}$. This restrictsto the bijection between the set $\tilt\Lambda$ of isomorphism classes of basictilting $\Lambda$-modules and the symmetric group $\mathfrak{S}_n$ due toBr\"{u}stle, Hille, Ringel and R\"{o}hrle. Regarding the preprojective algebra$\Gamma$ of Dynkin type $A_n$ as a factor algebra of $\Lambda$, we show thatthe tensor functor $-\otimes_{\Lambda}\Gamma$ induces a bijection between$\sttilt\Lambda\to\sttilt\Gamma$. This recover Mizuno's bijection$\mathfrak{S}_{n+1}\to\sttilt\Gamma$ for type $A_n$.

Classifying $\tau$-tilting modules over the Auslander algebra of $K[x]/(x^n)$