Patterns in rectangulations. Part I: $\mathrm{\top }$-like patterns, inversion sequence classes $I(010,101,120,201)$ and $I(011,201)$, and rushed Dyck paths

We initiate a systematic study of pattern avoidance in rectangulations. We give a formal definition of such patterns and investigate rectangulations that avoid $\mathrm{\top }$-like patterns - the pattern $\mathrm{\top }$ and its rotations. For every $L\subseteq \{\mathrm{\top },⊢,\mathrm{\perp },⊣\}$ we enumerate $L$-avoiding rectangulations, both weak and strong. In particular, we show $\mathrm{\top }$-avoiding weak rectangulations are enumerated by Catalan numbers and construct bijections to several Catalan structures. Then, we prove that $\mathrm{\top }$-avoiding strong rectangulations are in bijection with several classes of inversion sequences, among them $I(010,101,120,201)$ and $I(011,201)$ - which leads to a solution of the conjecture that these classes are Wilf-equivalent. Finally, we show that $\{\mathrm{\top },\mathrm{\perp }\}$-avoiding strong rectangulations are in bijection with recently introduced rushed Dyck paths.