Applications in Enumerative Combinatorics of Infinite Weighted Automata and Graphs

TL;DR

This paper introduces a unified methodology using infinite weighted automata, generating functions, and continued fractions to solve classical lattice path counting problems like Dyck and Motzkin paths.

Contribution

It presents a novel, general approach extending Rutten's method to a broader class of combinatorial enumeration problems.

Findings

Unified framework for lattice path enumeration

Application to Dyck and Motzkin path counting

Extension of existing automata-based methods

Abstract

In this paper we studied infinite weighted automata and a general methodology to solve a wide variety of classical lattice path counting problems in an uniform way. This counting problems are related to Dyck paths, Motzkin paths and some generalizations. These methodology uses weighted automata, equations of ordinary generating functions and continued fractions. It is a variation of the one proposed by J. Rutten.