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We classify $n$-hereditary monomial algebras in three natural contexts:First, we give a classification of the $n$-hereditary truncated path algebras.We show that they are exactly the $n$-representation-finite Nakayama algebrasclassified by Vaso. Next, we classify partially the $n$-hereditary quadraticmonomial algebras. In the case $n=2$, we prove that there are only twoexamples, provided that the preprojective algebra is a planar quiver withpotential. The first one is a Nakayama algebra and the second one is obtainedby mutating $\mathbb A_3\otimes_k \mathbb A_3$, where $\mathbb A_3$ is theDynkin quiver of type $A$ with bipartite orientation. In the case $n\geq 3$, weshow that the only $n$-representation finite algebras are the$n$-representation-finite Nakayama algebras with quadratic relations.

Classification results for $n$-hereditary monomial algebras