Replicated algebras derived equivalent to higher Auslander algebras of type A

We show that every higher Auslander algebra $A_{n+1}^d$ of type $\mathbb{A}$ such that $\gcd(n,d)=1$ is derived equivalent to a certain replicated algebra $B=B_0^{(n+d)}$. Moreover ${\rm{gldim}} B = nd$ and $B$ admits an $nd$-cluster tilting subcategory consisting of all direct sums of projective modules and injective modules. We introduce a class of algebras called $2$-subhomogeneous $m$-representation finite to characterize the homological properties of $B$ and give a method to obtain derived equivalences between fractionally Calabi-Yau algebras and $2$-subhomogeneous algebras using certain tilting complexes.