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Orders induced by segments in floorplan partitions and (2-14-3,3-41-2)-avoiding permutations

Andrei Asinowski, Gill Barequet, Mireille Bousquet-Mélou, Toufik Mansour, Ron Pinter

TL;DR

This work extends the Baxter-permutation framework from rectangles to floorplan segments by introducing S-equivalence and the S-permutation, establishing that S(P) avoids the pair $(2-14-3, 3-41-2)$ and provides a bijection with S-equivalence classes. It builds a bridge to the R-permutation theory, showing that R-equivalence refines S-equivalence and that the pair $(R(P), S(P))$ forms a complete Baxter permutation, enabling a unified view with complete Baxter structure. The paper also characterizes and enumerates these avoidance classes, presents linear-time construction methods to pass between permutations and floorplans, and analyzes the guillotine-floorplan special case via separable-by-point permutations and Schröder/Catalan-type enumerations. The results yield new enumerative formulas, highlight a deep connection between segment-based floorplan orderings and Baxter-type permutations, and provide a foundation for rectangulation problems and related geometric-combinatorial questions.

Abstract

A floorplan is a tiling of a rectangle by rectangles. There are natural ways to order the elements---rectangles and segments---of a floorplan. Ackerman, Barequet and Pinter studied a pair of orders induced by neighborhood relations between rectangles, and obtained a natural bijection between these pairs and (2-41-3, 3-14-2)-avoiding permutations, also known as (reduced) Baxter permutations. In the present paper, we first perform a similar study for a pair of orders induced by neighborhood relations between segments of a floorplan. We obtain a natural bijection between these pairs and another family of permutations, namely (2-14-3, 3-41-2)-avoiding permutations. Then, we investigate relations between the two kinds of pairs of orders---and, correspondingly, between (2-41-3, 3-14-2)- and (2-14-3, 3-41-2)-avoiding permutations. In particular, we prove that the superposition of both permutations gives a complete Baxter permutation (originally called w-admissible, by Baxter and Joichi in the sixties). In other words, (2-14-3, 3-41-2)-avoiding permutations are the hidden part of complete Baxter permutations. We enumerate these permutations. To our knowledge, the characterization of these permutations in terms of forbidden patterns and their enumeration are both new results. Finally, we also study the special case of the so-called guillotine floorplans.

Orders induced by segments in floorplan partitions and (2-14-3,3-41-2)-avoiding permutations

TL;DR

This work extends the Baxter-permutation framework from rectangles to floorplan segments by introducing S-equivalence and the S-permutation, establishing that S(P) avoids the pair and provides a bijection with S-equivalence classes. It builds a bridge to the R-permutation theory, showing that R-equivalence refines S-equivalence and that the pair forms a complete Baxter permutation, enabling a unified view with complete Baxter structure. The paper also characterizes and enumerates these avoidance classes, presents linear-time construction methods to pass between permutations and floorplans, and analyzes the guillotine-floorplan special case via separable-by-point permutations and Schröder/Catalan-type enumerations. The results yield new enumerative formulas, highlight a deep connection between segment-based floorplan orderings and Baxter-type permutations, and provide a foundation for rectangulation problems and related geometric-combinatorial questions.

Abstract

A floorplan is a tiling of a rectangle by rectangles. There are natural ways to order the elements---rectangles and segments---of a floorplan. Ackerman, Barequet and Pinter studied a pair of orders induced by neighborhood relations between rectangles, and obtained a natural bijection between these pairs and (2-41-3, 3-14-2)-avoiding permutations, also known as (reduced) Baxter permutations. In the present paper, we first perform a similar study for a pair of orders induced by neighborhood relations between segments of a floorplan. We obtain a natural bijection between these pairs and another family of permutations, namely (2-14-3, 3-41-2)-avoiding permutations. Then, we investigate relations between the two kinds of pairs of orders---and, correspondingly, between (2-41-3, 3-14-2)- and (2-14-3, 3-41-2)-avoiding permutations. In particular, we prove that the superposition of both permutations gives a complete Baxter permutation (originally called w-admissible, by Baxter and Joichi in the sixties). In other words, (2-14-3, 3-41-2)-avoiding permutations are the hidden part of complete Baxter permutations. We enumerate these permutations. To our knowledge, the characterization of these permutations in terms of forbidden patterns and their enumeration are both new results. Finally, we also study the special case of the so-called guillotine floorplans.

Paper Structure

This paper contains 22 sections, 22 theorems, 15 equations, 30 figures.

Table of Contents

  1. Introduction
  2. Orders between segments of a floorplan
  3. A bijection between S-equivalence classes of floorplans and $(2\hbox{-}14\hbox{-}3, 3\hbox{-}41\hbox{-}2)$-avoiding permutations
  4. S-equivalence
  5. S-permutations
  6. $(2\hbox{-}14\hbox{-}3, 3\hbox{-}41\hbox{-}2)$-Avoiding permutations
  7. S-permutations coincide with $(2\hbox{-} 14\hbox{-} 3, 3\hbox{-} 41\hbox{-} 2)$-avoiding permutations
  8. A. Directions of the segments $K_i$
  9. B. Endpoints of the segments $K_i$
  10. C. The construction indeed determines a floorplan
  11. D. For any floorplan $P$ obtained by the construction described above, $S(P)=\sigma$
  12. Relations between the R- and S-permutations
  13. Constructing $S(P)$ from $R(P)$
  14. Floorplans that produce the same S-permutation
  15. Enumeration of $(2\hbox{-}14\hbox{-}3, 3\hbox{-}41\hbox{-}2)$-avoiding permutations
  16. ...and 7 more sections

Key Result

Lemma 2.5

In the $\twoheadleftarrow$ order, $J$ covers $I$ if and only if $J$ is a right-neighbor of $I$. A similar statement holds for the $\twoheaddownarrow$ order.

Figures (30)

  • Figure 1: Two R-equivalent floorplans.
  • Figure 2: A rectangulation: embedding a point set $\Pi$ in a floorplan $P$.
  • Figure 3: Constructing the R-permutation of a floorplan $P$.
  • Figure 4: The segment $I$ is a left-neighbor of the segment $J$.
  • Figure 5: The right-neighbors of a vertical segment $I$ (thick segments). Note that the vertical segment $J$ is not a right-neighbor of $I$.
  • ...and 25 more figures

Theorems & Definitions (48)

  • Definition 2.1
  • Definition 2.2
  • proof
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • Definition 2.7
  • Proposition 2.8
  • proof
  • ...and 38 more