Length and area generating functions for height-restricted Motzkin meanders

TL;DR

This paper derives generating functions for height-restricted Motzkin meanders, providing explicit formulas and structural insights relevant to statistical mechanics models of physical systems.

Contribution

It introduces a novel embedding of Motzkin paths into anisotropic Dyck paths and develops explicit polynomial structures for their generating functions.

Findings

Derived length and area generating functions for Motzkin meanders.

Presented a cluster expansion revealing polynomial structure of generating functions.

Applied techniques relevant to physical models like polymers and interfaces.

Abstract

We derive the length and area generating function of planar height-restricted forward-moving discrete paths of increments +1, 0, or -1 with arbitrary starting and ending points, the so-called Motzkin meanders, and the more general length-area generating functions for Motzkin paths with markers monitoring the number of passages from the two height boundaries ('floor' and 'ceiling') and the time spent there. The results are obtained by embedding Motzkin paths in a two-step anisotropic Dyck path process and using propagator, exclusion statistics and bosonization techniques. We also present a cluster expansion of the logarithm of the generating functions that makes their polynomial structure explicit. These results are relevant to the derivation of statistical mechanical properties of physical systems such as polymers, vesicles, and solid-on-solid interfaces.