From dimer models to generalized lattice paths

TL;DR

This paper extends the connection between dimer models and lattice path generating functions to generalized paths like k-Dyck, k-Motzkin, and k-Schr"oder paths, introducing new combinatorial models and enumeration techniques.

Contribution

It introduces five new generalizations of the dimer model that relate to generalized lattice paths and derives their generating functions using recurrence relations and Lagrange inversion.

Findings

Derived recurrence relations for generalized lattice path generating functions

Expressed generating functions in terms of generalized paths with weights

Enumerated paths considering statistics like size, area, peaks, valleys, and heights

Abstract

A recurrence relation of the generating function of the dimer model of Fibonacci type gives a functional relation for formal power series associated to lattice paths such as a Dyck, Motzkin and Schr\"oder path. In this paper, we generalize the correspondence to the case of generalized lattice paths, $k$-Dyck, $k$-Motzkin and $k$-Schr\"oder paths, by modifying the recurrence relation of the dimer model. We introduce five types of generalizations of the dimer model by keeping its combinatorial structures. This allows us to express the generating functions in terms of generalized lattice paths. The weight given to a generalized lattice path involves several statistics such as size, area, peaks and valleys, and heights of horizontal steps. We enumerate the generalized lattice paths by use of the recurrence relations and the Lagrange inversion theorem.