Nakayama Algebras which are Higher Auslander Algebras

We prove that any cyclic Nakayama algebra which is a higher Auslander algebra can be uniquely constructed from Nakayama algebras of smaller ranks by reversing the syzygy filtration process. This creates chains of higher Auslander algebras upto $\boldsymbol\varepsilon$-equivalences. Therefore, the classification of all cyclic Nakayama algebras which are higher Auslander algebras reduces to the classification of linear ones. We give two applications of this: for any integer $k$ where $2\leq k\leq 2n-2$, there is a Nakayama algebra of rank $n$ which is a higher Auslander algebra of global dimension $k$ and the possible values of the global dimensions of cyclic Nakayama algebras which are higher Auslander algebras form the sets $\left\{2,\ldots,2n-2\right\}\setminus\left\{n-1\right\}$ if $n$ is even and $\left\{2,\ldots,2n-2\right\}\setminus\left\{ 2,n-1\right\}$ if $n$ is odd.