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Higher dimensional floorplans and Baxter d-permutations

Nicolas Bonichon, Thomas Muller, Adrian Tanasa

TL;DR

This work extends mosaic floorplan theory to $d$ dimensions, introducing $d$-floorplans and their associated $d$-permutations. It develops a generating-tree framework based on block deletions and pushable corners, enabling efficient enumeration and structural understanding of $d$-floorplans via a vector-label encoding. A central result is a bijection between $2^{d-1}$-floorplans and a subclass of $d$-permutations defined by forbidden vincular patterns, with $ ext{psi}=\phi^{-1}$, generalizing the classical mosaic floorplan–Baxter permutation correspondence. The bijection links higher-dimensional guillotine-like partitions and separable $d$-permutations, situating the new objects within the broader landscape of pattern-avoiding combinatorics and providing a concrete framework for exploring higher-dimensional floorplans and their combinatorial enumerations.

Abstract

A $2-$dimensional mosaic floorplan is a partition of a rectangle by other rectangles with no empty rooms. These partitions (considered up to some deformations) are known to be in bijection with Baxter permutations. A $d$-floorplan is the generalisation of mosaic floorplans in higher dimensions, and a $d$-permutation is a $(d-1)$-tuple of permutations. Recently, in N. Bonichon and P.-J. Morel, {\it J. Integer Sequences} 25 (2022), Baxter $d$-permutations generalising the usual Baxter permutations were introduced. In this paper, we consider mosaic floorplans in arbitrary dimensions, and we construct a generating tree for $d$-floorplans, which generalises the known generating tree structure for $2$-floorplans. The corresponding labels and rewriting rules appear to be significantly more involved in higher dimensions. Moreover we give a bijection between the $2^{d-1}$-floorplans and $d$-permutations characterized by forbidden vincular patterns. Surprisingly, this set of $d$-permutations is strictly contained within the set of Baxter $d$-permutations.

Higher dimensional floorplans and Baxter d-permutations

TL;DR

This work extends mosaic floorplan theory to dimensions, introducing -floorplans and their associated -permutations. It develops a generating-tree framework based on block deletions and pushable corners, enabling efficient enumeration and structural understanding of -floorplans via a vector-label encoding. A central result is a bijection between -floorplans and a subclass of -permutations defined by forbidden vincular patterns, with , generalizing the classical mosaic floorplan–Baxter permutation correspondence. The bijection links higher-dimensional guillotine-like partitions and separable -permutations, situating the new objects within the broader landscape of pattern-avoiding combinatorics and providing a concrete framework for exploring higher-dimensional floorplans and their combinatorial enumerations.

Abstract

A dimensional mosaic floorplan is a partition of a rectangle by other rectangles with no empty rooms. These partitions (considered up to some deformations) are known to be in bijection with Baxter permutations. A -floorplan is the generalisation of mosaic floorplans in higher dimensions, and a -permutation is a -tuple of permutations. Recently, in N. Bonichon and P.-J. Morel, {\it J. Integer Sequences} 25 (2022), Baxter -permutations generalising the usual Baxter permutations were introduced. In this paper, we consider mosaic floorplans in arbitrary dimensions, and we construct a generating tree for -floorplans, which generalises the known generating tree structure for -floorplans. The corresponding labels and rewriting rules appear to be significantly more involved in higher dimensions. Moreover we give a bijection between the -floorplans and -permutations characterized by forbidden vincular patterns. Surprisingly, this set of -permutations is strictly contained within the set of Baxter -permutations.

Paper Structure

This paper contains 15 sections, 12 theorems, 21 equations, 21 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries and definitions
  3. Generalisation of rectangles in dimension d
  4. Mosaic floorplans in dimension d
  5. Equivalence relation for d-floorplans
  6. Block deletion d-floorplans
  7. A Generating tree of $d$-floorplans
  8. Pushable corners and block insertion
  9. Pushable corners of the children of a $d$-floorplan
  10. Encoding the generating tree
  11. Bijection between higher dimensional floorplans and d-permutations
  12. Definition of the bijection
  13. Forbidden pattern and main theorem
  14. Proof of the bijection
  15. Proof of theorem \ref{['th:final']} :

Key Result

Lemma 2.1

Let $B$ be a block of a $d$-floorplan ${\mathcal{P}}$ and $q$ corner of $B$ that is not on the boundary of ${\mathcal{P}}$. The relation touch (see Definition def:touch) is a total order on $b(\mathcal{F})_{B,q}$.

Figures (21)

  • Figure 1: On the left an example of a $3-$floorplan. In the middle the relative order of each blocks with respect to each direction ($x,y,z$). On the right the corresponding $3$-permutation (considering the $3$-floorplan as a $4$-floorplan).
  • Figure 2: On the left two facets crossing. In the middle, facet $B$ touches facet $A$. On the right, two facets touching each other.
  • Figure 3: A non-generic 3-floorplan. On the left the boxes. On the right the borders with "L" shapes.
  • Figure 4: The first 4-floorplans.
  • Figure 6: Examples of $3-$floorplans
  • ...and 16 more figures

Theorems & Definitions (49)

  • Definition 1
  • Example 1
  • Definition 2
  • Definition 3
  • Example 2
  • Definition 4
  • Example 3
  • Remark 5
  • Definition 6
  • Example 4
  • ...and 39 more