Patterns in rectangulations. Part I: -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 -like patterns - the pattern and its rotations. For every L ⊆ \, \, , \, , \, \ we enumerate L-avoiding rectangulations, both weak and strong. In particular, we show -avoiding weak rectangulations are enumerated by Catalan numbers and construct bijections to several Catalan structures. Then, we prove that -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 \, \-avoiding strong rectangulations are in bijection with recently introduced rushed Dyck paths. 30 pages, 21 figures