Generic rectangulations

A rectangulation is a tiling of a rectangle by a finite number of rectangles. The rectangulation is called generic if no four of its rectangles share a single corner. We initiate the enumeration of generic rectangulations up to combinatorial equivalence by establishing an explicit bijection between generic rectangulations and a set of permutations defined by a pattern-avoidance condition analogous to the definition of the twisted Baxter permutations.

💡 Research Summary

The paper studies generic rectangulations, which are tilings of a fixed rectangle by a finite number of smaller rectangles such that no four rectangles meet at a single interior corner (a “cross”). Two tilings are considered combinatorially equivalent if there is a bijection between their constituent rectangles preserving the “below” and “left of” relations. The main goal is to enumerate generic rectangulations up to this equivalence.

To achieve this, the author introduces a new family of permutations called k‑clumped permutations. For a permutation x, each descent x_i > x_{i+1} determines a set of values lying strictly between x_i and x_{i+1}; these values form a clump if they all lie on the same side of the two entries. A permutation is k‑clumped if every descent has at most k clumps. The case k = 1 coincides with the well‑known twisted Baxter permutations (which avoid the patterns 2‑41‑3 and 3‑41‑2). The paper focuses on k = 2. A 2‑clumped permutation avoids the four generalized patterns 3‑51‑2‑4, 3‑51‑4‑2, 2‑4‑51‑3, and 4‑2‑51‑3. The set of all 2‑clumped permutations of size n is denoted G_n.

The author then connects these permutations to the weak order on S_n, the lattice of permutations ordered by inclusion of inversion sets. Using the framework of lattice congruences from Reading’s earlier work, a specific congruence Γ on the weak order is defined via a collection C = {35124, 24513} of untranslated join‑irreducible permutations. Theorem 2.1 (adapted from Reading’s Theorem 9.3) shows that the minimal elements of each Γ‑class are exactly the 2‑clumped permutations, and that two permutations lie in the same class precisely when a certain pattern (an “adjacent cliff”) appears with entries satisfying a specific interleaving condition.

Next, the paper reviews diagonal rectangulations, a subclass of generic rectangulations where every rectangle’s interior intersects the main diagonal of the outer rectangle. A known map ρ : S_n → DiagonalRect_n, originally described by Reading and later by others, constructs a diagonal rectangulation from a permutation by inserting rectangles one by one according to the order of the permutation’s entries on the diagonal. The map ρ is surjective, and its fibers consist of all permutations compatible with the resulting diagonal rectangulation; compatibility means that after each insertion the union of already placed rectangles is left‑ and bottom‑justified.

The central construction of the paper is a map γ : S_n → gRec_n (the set of all generic rectangulations). Given a permutation x, one first builds the diagonal rectangulation ρ(x). Then, using wall slides—local moves that slide two adjacent walls past each other without altering the rest of the structure—one generates all generic rectangulations that are mosaic floorplans equivalent to ρ(x). The crucial observation is that the fibers of γ coincide exactly with the Γ‑congruence classes on S_n. Consequently, restricting γ to the set of minimal elements of each class (i.e., the 2‑clumped permutations) yields a bijection \