Combinatorics of rectangulations: Old and new bijections

TL;DR

The paper develops a uniform framework linking rectangulations to permutation classes through weak and strong equivalence relations, providing constructive mappings γ_w and γ_s that realize fibers as linear extensions of adjacency-type posets. It cleanly separates weak and strong theories, showing that weak rectangulations correspond to Baxter-type families and that strong rectangulations correspond to 2-clumped/co-2-clumped permutations, with diagonal and NW–SE/SW–NE labelings underpinning the bijections. A key advance is the guillotine case: windmills are encoded by two mesh patterns, yielding new permutation-class correspondences and enabling enumeration; the authors also build a quadrant-walk framework to count strong rectangulations and derive asymptotic bounds. Overall, the work unifies prior results, introduces simpler forward/backward algorithms, and provides new combinatorial and enumerative insights into rectangulations and their permutation encodings with practical implications for related lattice and polytope structures via quotient constructions.

Abstract

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.

Figures (31)

• Figure 1: Artwork Composition décentralisée (1924) by Theo van Doesburg (Dutch, 1883--1931). Solomon R. Guggenheim Museum, New York.
• Figure 2: Illustration to Observation \ref{['obs:lrab']}: four regions delimited by alternating paths.
• Figure 3: $\mathcal{R}_1$, $\mathcal{R}_2$, and $\mathcal{R}_3$ are weakly equivalent. $\mathcal{R}_1$ and $\mathcal{R}_2$ are strongly equivalent. $\mathcal{R}_4$ is guillotine.
• Figure 4: The rectangles $r_i$ with $i>j$ (respectively $i<j$) are located to the right or below $r_j$ (respectively to the left or above $r_j$).
• Figure 5: The NW--SE (left) and the SW--NE (right) orderings of the rectangles of $\mathcal{R}_1$.

Theorems & Definitions (51)

• Proposition 2
• Proposition 3
• Proposition 4
• Proposition 5
• Theorem 6
• Theorem 7
• Proposition 8
• proof
• Proposition 9
• proof