Type A Nilpotent Hessenberg varieties with Hessenberg function $h(i)\le i+1$

We study the type A nilpotent Hessenberg varieties associated with Hessenberg functions that satisfy $h(i)\le i+1$. We call these the generalized parabolic Peterson varieties. We show that such varieties can be decomposed into the union of specific generalized parabolic Peterson varieties $\operatorname{Pet}_{\lambda,\alpha}$, such that $\lambda$ is an integer partition, $\alpha$ is an integer composition, and $\alpha$ is dominated by $\lambda$. We prove that when $\alpha$ is dominated by $\lambda$, the cardinality of the maximal dimensional components of $\operatorname{Pet}_{\lambda,\alpha}$ equals the Kostka number $\mathcal{K}_{\lambda\alpha}$, and its dimension is determined only by $\lambda$ and the length of $\alpha$. We provide a recursive formula for the Poincar\'e polynomial of $\operatorname{Pet}_{\lambda,\alpha}$. These results partially answer open questions about the geometry of Hessenberg varieties but raise further questions about the representation-theoretic reasons for these facts.