A rectangulation is a decomposition of a rectangle into finitely many rectangles. Via natural equivalence relations, rectangulations can be seen as combinatorial objects with a rich structure, with links to lattice congruences, flip graphs, polytopes, lattice paths, Hopf algebras, etc. In this paper, we first revisit the structure of the respective equivalence classes: weak rectangulations that preserve rectangle-segment adjacencies, and strong rectangulations that preserve rectangle-rectangle adjacencies. We thoroughly investigate posets defined by adjacency in rectangulations of both kinds, and unify and simplify known bijections between rectangulations and permutation classes. This yields a uniform treatment of mappings between permutations and rectangulations that unifies the results from earlier contributions, and emphasizes parallelism and differences between the weak and the strong cases. Then, we consider the special case of guillotine rectangulations, and prove that they can be characterized - under all known mappings between permutations and rectangulations - by avoidance of two mesh patterns that correspond to "windmills" in rectangulations. This yields new permutation classes in bijection with weak guillotine rectangulations, and the first known permutation class in bijection with strong guillotine rectangulations. Finally, we address enumerative issues and prove asymptotic bounds for several families of strong rectangulations.

翻译：矩形剖分是将一个矩形分解为有限多个矩形。通过自然等价关系，矩形剖分可被视为具有丰富结构的组合对象，与格同余、翻转图、多面体、格路、Hopf代数等存在联系。本文首先重新审视了各类等价类的结构：弱矩形剖分保留矩形-线段邻接关系，强矩形剖分保留矩形-矩形邻接关系。我们深入研究了两种矩形剖分中由邻接关系定义的偏序集，统一并简化了矩形剖分与置换类之间已知的双射。这为置换与矩形剖分之间的映射提供了统一处理方法，既整合了早期成果中的结论，又强调了弱情形与强情形之间的平行性与差异性。随后，我们考虑断头台矩形剖分的特例，证明在所有已知的置换与矩形剖分映射下，该剖分可由避免两种对应于矩形剖分中"风车"的网格模式来刻画。这产生了与弱断头台矩形剖分双射的新置换类，以及首个与强断头台矩形剖分双射的已知置换类。最后，我们探讨了计数问题，并给出了若干强矩形剖分族的渐近界。