Lattice paths with a first return decomposition constrained by the maximal height of a pattern

TL;DR

This paper develops a method to find closed-form solutions for generating functions of lattice paths constrained by height and pattern conditions, applying it to various path classes like Dyck and Motzkin paths.

Contribution

It introduces a general approach to solve recursive equations for path enumeration with height constraints and applies it to multiple path families with pattern-based restrictions.

Findings

Derived closed-form expressions for generating functions of constrained lattice paths.

Extended enumeration techniques to Dyck, Motzkin, skew Dyck, and skew Motzkin paths.

Demonstrated the applicability of the method to various pattern-restricted path classes.

Abstract

We consider the system of equations $A_k(x)=p(x)A_{k-1}(x)(q(x)+\sum_{i=0}^k A_i(x))$ for $k\geq r+1$ where $A_i(x)$, $0\leq i \leq r$, are some given functions and show how to obtain a close form for $A(x)=\sum_{k\geq 0}A_k(x)$. We apply this general result to the enumeration of certain subsets of Dyck, Motzkin, skew Dyck, and skew Motzkin paths, defined recursively according to the first return decomposition with a monotonically non-increasing condition relative to the maximal ordinate reached by an occurrence of a given pattern $\pi$.