Some classes of sequences of Linear Type

Given a graded ring $A$ and a homogeneous ideal $I$, the ideal is said to be of linear type if the Rees algebra of $I$ is isomorphic to the symmetric algebra of $I$. In general, $y$-regularity of Rees algebra of $I$ is $0 \Rightarrow$ $I$ is generated by a $d$-sequence $\Rightarrow I$ is of linear type. We show that $d$-sequence ideals represent a significantly smaller subset of ideals of linear type in terms of $y$-regularity. Moreover, we identify a class of $d$-sequences whose arbitrary powers generate ideals of Gr\"obner linear type. Notably, while $d$-sequences are inherently weak $d$-sequences, we highlight a specific class of algebras where weak $d$-sequences are indeed $d$-sequences.