Enumerating Pattern-Avoiding Rectangulations and Their Connections to Permutations

المفاهيم الأساسية

The article establishes new bijective links between pattern-avoiding rectangulations and pattern-avoiding permutations, proving that their generating functions are algebraic.

الملخص

The article focuses on enumerating various classes of pattern-avoiding rectangulations and exploring their connections to pattern-avoiding permutations.
Key highlights:

Rectangulations are tilings of a rectangle by smaller rectangles, and they can be classified based on the patterns they avoid.
The authors establish bijections between certain classes of pattern-avoiding rectangulations and pattern-avoiding permutations.
They prove that the generating functions for these classes are algebraic, confirming several conjectures by Merino and Mütze.
The authors also analyze a new class of rectangulations called "whirls" using a generating tree approach and provide a closed-form expression for their generating function.
The results demonstrate the deep connections between geometric structures like rectangulations and combinatorial objects like permutations, leading to insights about their enumeration and properties.

الإحصائيات

We enumerate several classes of pattern-avoiding rectangulations. We establish new bijective links with pattern-avoiding permutations, prove that their generating functions are algebraic, and confirm several conjectures by Merino and Mütze.
Rectangulations of size n are tilings of a rectangle by n rectangles such that no four rectangles meet in a point.

اقتباسات

"Rectangulations are also called floorplans or rectangular dissections."
"Such structures appear naturally for architectural building plans, integrated circuits (see Figure 1), and were investigated since the 70s with some graph theory, computational geometry, and combinatorial optimization point of views."
"It was shown that some important families of rectangulations are enumerated by famous integer sequences (e.g., Baxter, Schröder, Catalan numbers) and that they have strong links with pattern-avoiding permutations."

الرؤى الأساسية المستخلصة من

From geometry to generating functions

by Andrei Asino... في arxiv.org 04-02-2024

https://arxiv.org/pdf/2401.05558.pdf

استفسارات أعمق

What other geometric structures or combinatorial objects could be explored in a similar manner to uncover new connections and insights?

In a similar manner to the exploration of rectangulations and permutations, other geometric structures and combinatorial objects could be investigated to reveal new connections and insights. One potential area of study could be polygon dissections, where polygons are divided into smaller polygons with specific constraints. By applying bijections and generating functions, researchers could uncover relationships between different types of dissections and potentially link them to other combinatorial objects or sequences. Additionally, exploring lattice paths or polyomino tilings using similar techniques could lead to the discovery of interesting patterns and connections within these structures.

How might the techniques used in this article, such as bijections and generating functions, be applied to study other classes of pattern-avoiding structures beyond rectangulations and permutations?

The techniques of bijections and generating functions showcased in the article can be effectively applied to study various other classes of pattern-avoiding structures beyond rectangulations and permutations. For instance, these methods could be utilized to analyze pattern-avoiding polyominoes, where certain configurations of connected squares are prohibited. By establishing bijections between polyomino tilings and permutations, researchers can enumerate and analyze the properties of pattern-avoiding polyomino structures. Generating functions can then be employed to derive algebraic expressions for the counting sequences of these structures, providing valuable insights into their combinatorial properties.

Are there any potential applications or practical implications of the results presented in this article, for example in the fields of architecture, integrated circuit design, or combinatorial optimization?

The results presented in the article on rectangulations and permutations have potential applications and practical implications in various fields. In architecture, the study of rectangulations can offer insights into efficient floorplan designs and optimal space utilization in building layouts. Integrated circuit design can benefit from the combinatorial optimization techniques used to analyze pattern-avoiding structures, leading to improved circuit layouts and minimized signal interference. Additionally, the findings could be applied in combinatorial optimization problems in logistics, scheduling, and network design, where efficient arrangements and configurations are crucial for optimal performance. By leveraging the results from this research, practitioners in these fields can enhance their design processes and decision-making strategies.