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Parameterizing V-notch Weir Equations for Flow Monitoring in a Drainage Control Structure

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Published on: April 25, 2025

Length and area generating functions for height-restricted Motzkin meanders.

Alexios P Polychronakos¹

¹Physics Department, the City College of New York, New York 10031, USA and The Graduate Center, CUNY, New York, New York 10016, USA.

Physical Review. E

|March 16, 2022

Summary

This study introduces generating functions for Motzkin meanders and paths, detailing their length and area. These findings are crucial for understanding statistical mechanics in physical systems like polymers.

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Area of Science:

Combinatorics
Statistical Physics
Mathematical Physics

Background:

Discrete paths, such as Motzkin paths, are fundamental in combinatorics.
Understanding their statistical properties is key to modeling physical systems.
Previous work has focused on simpler path properties, leaving complex generating functions unexplored.

Purpose of the Study:

Derive length and area generating functions for Motzkin meanders and generalized Motzkin paths.
Incorporate markers for boundary passages and time spent at boundaries.
Provide a framework for statistical mechanical analysis of related physical systems.

Main Methods:

Embedding Motzkin paths within a two-step anisotropic Dyck path process.
Utilizing propagator, exclusion statistics, and bosonization techniques.
Applying a cluster expansion to reveal the polynomial structure of generating functions.

Main Results:

Derived the length and area generating functions for Motzkin meanders.
Developed generalized generating functions for Motzkin paths with boundary passage markers.
Exhibited the polynomial structure of these generating functions via cluster expansion.

Conclusions:

The derived generating functions offer a powerful tool for analyzing Motzkin paths.
These results facilitate the study of statistical mechanical properties of polymers, vesicles, and interfaces.
The methods employed provide a novel approach to complex combinatorial problems.