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Analyzing Scaling of Random Tamari Intervals and Schnyder Woods in Triangulations


Core Concepts
Height of random Tamari intervals scales as n^3/4 with an explicit limit law.
Abstract
The content discusses the scaling of random Tamari intervals and Schnyder woods in random triangulations. It delves into the exact solutions of models based on polynomial equations and D-finite tricks. The article explores the convergence of moments and the universality of the scaling phenomenon. Key insights include the bijection between intervals and triangulations, the role of the Tamari lattice in algebraic combinatorics, and the application of the D-finite method for moment pumping.
Stats
The height of a uniformly chosen vertex scales as n^3/4. The number of elements in Tamari intervals is given by a specific formula. The exact solution is based on polynomial equations with catalytic variables.
Quotes
"We show that the height of a uniformly chosen vertex scales as n^3/4."

Deeper Inquiries

What implications does the convergence of moments have in other combinatorial models

The convergence of moments in combinatorial models has significant implications for understanding the behavior of random structures. By studying the scaling of moments, researchers can gain insights into the distribution of random variables in these models. This information can be used to make predictions about the properties of large random structures, such as the heights of points in decomposition trees or the sizes of certain components in combinatorial objects. Understanding the convergence of moments allows for the analysis of the limiting behavior of these structures as they grow in size, providing valuable information for various applications in combinatorics and probability theory.

How does the universality of the scaling phenomenon impact the study of decomposition trees

The universality of the scaling phenomenon in decomposition trees suggests that certain scaling behaviors are common across different combinatorial models. This universality implies that the convergence of moments and the scaling limits observed in one model may also apply to other related models. By identifying and studying these universal scaling properties, researchers can gain a deeper understanding of the underlying structures and relationships between different combinatorial objects. This can lead to the development of general theories and techniques that apply to a wide range of combinatorial problems, enhancing the overall understanding of these complex systems.

What are the practical applications of the D-finite method beyond this specific context

The D-finite method, as demonstrated in the context of the convergence of moments in combinatorial models, has practical applications beyond this specific scenario. This method provides a systematic approach to solving functional equations with catalytic variables, allowing for the explicit calculation of generating functions and the analysis of their properties. By applying the D-finite method, researchers can efficiently study the asymptotic behavior of combinatorial structures, derive explicit formulas for generating functions, and analyze the convergence of moments. This method can be utilized in various combinatorial problems to obtain precise results and insights into the underlying structures and their scaling properties.
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